Univariate Polynomial Base Class

Univariate Polynomial Base Class

AUTHORS:

  • William Stein: first version.
  • Martin Albrecht: Added singular coercion.
  • Robert Bradshaw: Move Polynomial_generic_dense to Cython.
  • Miguel Marco: Implemented resultant in the case where PARI fails.
  • Simon King: Use a faster way of conversion from the base ring.
  • Julian Rueth (2012-05-25,2014-05-09): Fixed is_squarefree() for imperfect fields, fixed division without remainder over QQbar; added _cache_key for polynomials with unhashable coefficients
  • Simon King (2013-10): Implement copying of PolynomialBaseringInjection.

TESTS:

sage: R.<x> = ZZ[]
sage: f = x^5 + 2*x^2 + (-1)
sage: f == loads(dumps(f))
True
class sage.rings.polynomial.polynomial_element.ConstantPolynomialSection

Bases: sage.categories.map.Map

This class is used for conversion from a polynomial ring to its base ring.

Since trac ticket #9944, it calls the constant_coefficient method, which can be optimized for a particular polynomial type.

EXAMPLES:

sage: P0.<y_1> = GF(3)[]
sage: P1.<y_2,y_1,y_0> = GF(3)[]
sage: P0(-y_1)    # indirect doctest
2*y_1

sage: phi = GF(3).convert_map_from(P0); phi
Generic map:
  From: Univariate Polynomial Ring in y_1 over Finite Field of size 3
  To:   Finite Field of size 3
sage: type(phi)
<type 'sage.rings.polynomial.polynomial_element.ConstantPolynomialSection'>
sage: phi(P0.one_element())
1
sage: phi(y_1)
Traceback (most recent call last):
...
TypeError: not a constant polynomial
class sage.rings.polynomial.polynomial_element.Polynomial

Bases: sage.structure.element.CommutativeAlgebraElement

A polynomial.

EXAMPLE:

sage: R.<y> = QQ['y']
sage: S.<x> = R['x']
sage: f = x*y; f
y*x
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: p = (y+1)^10; p(1)
1024
add_bigoh(prec)

Returns the power series of precision at most prec got by adding \(O(q^\text{prec})\) to self, where q is its variable.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: f = 1 + 4*x + x^3
sage: f.add_bigoh(7)
1 + 4*x + x^3 + O(x^7)
sage: f.add_bigoh(2)
1 + 4*x + O(x^2)
sage: f.add_bigoh(2).parent()
Power Series Ring in x over Integer Ring
any_root(ring=None, degree=None, assume_squarefree=False)

Return a root of this polynomial in the given ring.

INPUT:

  • ring – The ring in which a root is sought. By default this is the coefficient ring.
  • degree (None or nonzero integer) – Used for polynomials over finite fields. Returns a root of degree abs(degree) over the ground field. If negative, also assumes that all factors of this polynomial are of degree abs(degree). If None, returns a root of minimal degree contained within the given ring.
  • assume_squarefree (bool) – Used for polynomials over finite fields. If True, this polynomial is assumed to be squarefree.

EXAMPLES:

sage: R.<x> = GF(11)[]
sage: f = 7*x^7 + 8*x^6 + 4*x^5 + x^4 + 6*x^3 + 10*x^2 + 8*x + 5
sage: f.any_root()
2
sage: f.factor()
(7) * (x + 9) * (x^6 + 10*x^4 + 6*x^3 + 5*x^2 + 2*x + 2)
sage: f = x^6 + 10*x^4 + 6*x^3 + 5*x^2 + 2*x + 2
sage: f.any_root(GF(11^6, 'a'))
a^5 + a^4 + 7*a^3 + 2*a^2 + 10*a
sage: sorted(f.roots(GF(11^6, 'a')))
[(10*a^5 + 2*a^4 + 8*a^3 + 9*a^2 + a, 1), (10*a^5 + 3*a^4 + 8*a^3 + a^2 + 10*a + 4, 1), (2*a^5 + 8*a^4 + 3*a^3 + 6*a + 2, 1), (9*a^5 + 5*a^4 + 10*a^3 + 8*a^2 + 3*a + 1, 1), (a^5 + 3*a^4 + 8*a^3 + 2*a^2 + 3*a + 4, 1), (a^5 + a^4 + 7*a^3 + 2*a^2 + 10*a, 1)]
sage: f.any_root(GF(11^6, 'a'))
a^5 + a^4 + 7*a^3 + 2*a^2 + 10*a

sage: g = (x-1)*(x^2 + 3*x + 9) * (x^5 + 5*x^4 + 8*x^3 + 5*x^2 + 3*x + 5)
sage: g.any_root(ring=GF(11^10, 'b'), degree=1)
1
sage: g.any_root(ring=GF(11^10, 'b'), degree=2)
6*b^9 + 7*b^7 + 7*b^6 + 3*b^5 + b^2 + b + 3
sage: g.any_root(ring=GF(11^10, 'b'), degree=5)
7*b^9 + 2*b^8 + 6*b^7 + 3*b^6 + 2*b^5 + 7*b^3 + 7*b^2 + 2

TESTS:

sage: R.<x> = GF(5)[]
sage: K.<a> = GF(5^12)
sage: for _ in range(40):
....:     f = R.random_element(degree=4)
....:     assert f(f.any_root(K)) == 0

Check that our Cantor-Zassenhaus implementation does not loop over finite fields of even characteristic (see trac ticket #16162):

sage: K.<a> = GF(2**8)
sage: x = polygen(K)
sage: (x**2+x+1).any_root() # used to loop
Traceback (most recent call last):
...
ValueError: no roots A 1
sage: (x**2+a+1).any_root()
a^7 + a^2

Also check that such computations can be interrupted:

sage: K.<a> = GF(2**8)
sage: x = polygen(K)
sage: alarm(1)
sage: (x**1000000+x+a).any_root()
Traceback (most recent call last):
...
AlarmInterrupt

Check root computation over large finite fields:

sage: K.<a> = GF(2**50)
sage: x = polygen(K)
sage: (x**10+x+a).any_root()
a^49 + a^47 + a^44 + a^42 + a^41 + a^39 + a^38 + a^37 + a^36 + a^34 + a^33 + a^29 + a^27 + a^26 + a^25 + a^23 + a^18 + a^13 + a^7 + a^5 + a^4 + a^3 + a^2 + a
sage: K.<a> = GF(2**150)
sage: x = polygen(K)
sage: (x**10+x+a).any_root()
a^149 + a^148 + a^146 + a^144 + a^143 + a^140 + a^138 + a^136 + a^134 + a^132 + a^131 + a^130 + a^129 + a^127 + a^123 + a^120 + a^118 + a^114 + a^113 + a^112 + a^111 + a^108 + a^104 + a^103 + a^102 + a^99 + a^98 + a^94 + a^91 + a^90 + a^88 + a^79 + a^78 + a^75 + a^73 + a^72 + a^67 + a^65 + a^64 + a^63 + a^62 + a^61 + a^59 + a^57 + a^52 + a^50 + a^48 + a^47 + a^46 + a^45 + a^43 + a^41 + a^39 + a^37 + a^34 + a^31 + a^29 + a^27 + a^25 + a^23 + a^22 + a^20 + a^18 + a^16 + a^14 + a^11 + a^10 + a^8 + a^6 + a^5 + a^4 + a + 1
args()

Returns the generator of this polynomial ring, which is the (only) argument used when calling self.

EXAMPLES:

sage: R.<x> = QQ[]
sage: x.args()
(x,)

A constant polynomial has no variables, but still takes a single argument.

sage: R(2).args()
(x,)
base_extend(R)

Return a copy of this polynomial but with coefficients in R, if there is a natural map from coefficient ring of self to R.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 - 17*x + 3
sage: f.base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no such base extension
sage: f.change_ring(GF(7))
x^3 + 4*x + 3
base_ring()

Return the base ring of the parent of self.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: x.base_ring()
Integer Ring
sage: (2*x+3).base_ring()
Integer Ring
change_ring(R)

Return a copy of this polynomial but with coefficients in R, if at all possible.

EXAMPLES:

sage: K.<z> = CyclotomicField(3)
sage: f = K.defining_polynomial()
sage: f.change_ring(GF(7))
x^2 + x + 1
change_variable_name(var)

Return a new polynomial over the same base ring but in a different variable.

EXAMPLES:

sage: x = polygen(QQ,'x')
sage: f = -2/7*x^3 + (2/3)*x - 19/993; f
-2/7*x^3 + 2/3*x - 19/993
sage: f.change_variable_name('theta')
-2/7*theta^3 + 2/3*theta - 19/993
coefficients()

Return the coefficients of the monomials appearing in self.

EXAMPLES:

sage: _.<x> = PolynomialRing(ZZ)
sage: f = x^4+2*x^2+1
sage: f.coefficients()
[1, 2, 1]
coeffs()

Returns self.list().

(It is potentially slightly faster to use self.list() directly.)

EXAMPLES:

sage: x = QQ['x'].0
sage: f = 10*x^3 + 5*x + 2/17
sage: f.coeffs()
[2/17, 5, 0, 10]
complex_roots()

Return the complex roots of this polynomial, without multiplicities.

Calls self.roots(ring=CC), unless this is a polynomial with floating-point coefficients, in which case it is uses the appropriate precision from the input coefficients.

EXAMPLES:

sage: x = polygen(ZZ)
sage: (x^3 - 1).complex_roots()   # note: low order bits slightly different on ppc.
[1.00000000000000, -0.500000000000000 - 0.86602540378443...*I, -0.500000000000000 + 0.86602540378443...*I]

TESTS:

sage: x = polygen(RR)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 53 bits of precision
sage: x = polygen(RDF)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Double Field
sage: x = polygen(RealField(200))
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 200 bits of precision
sage: x = polygen(CDF)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Double Field
sage: x = polygen(ComplexField(200))
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 200 bits of precision
sage: x=polygen(ZZ,'x'); v=(x^2-x-1).complex_roots()
sage: v[0].parent() is CC
True
constant_coefficient()

Return the constant coefficient of this polynomial.

OUTPUT: element of base ring

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = -2*x^3 + 2*x - 1/3
sage: f.constant_coefficient()
-1/3
content()

Return the content of self, which is the ideal generated by the coefficients of self.

EXAMPLES:

sage: R.<x> = IntegerModRing(4)[]
sage: f = x^4 + 3*x^2 + 2
sage: f.content()
Ideal (2, 3, 1) of Ring of integers modulo 4
degree(gen=None)

Return the degree of this polynomial. The zero polynomial has degree -1.

EXAMPLES:

sage: x = ZZ['x'].0
sage: f = x^93 + 2*x + 1
sage: f.degree()
93
sage: x = PolynomialRing(QQ, 'x', sparse=True).0
sage: f = x^100000
sage: f.degree()
100000
sage: x = QQ['x'].0
sage: f = 2006*x^2006 - x^2 + 3
sage: f.degree()
2006
sage: f = 0*x
sage: f.degree()
-1
sage: f = x + 33
sage: f.degree()
1

AUTHORS:

  • Naqi Jaffery (2006-01-24): examples
denominator()

Return a denominator of self.

First, the lcm of the denominators of the entries of self is computed and returned. If this computation fails, the unit of the parent of self is returned.

Note that some subclasses may implement their own denominator function. For example, see sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint

Warning

This is not the denominator of the rational function defined by self, which would always be 1 since self is a polynomial.

EXAMPLES:

First we compute the denominator of a polynomial with integer coefficients, which is of course 1.

sage: R.<x> = ZZ[]
sage: f = x^3 + 17*x + 1
sage: f.denominator()
1

Next we compute the denominator of a polynomial with rational coefficients.

sage: R.<x> = PolynomialRing(QQ)
sage: f = (1/17)*x^19 - (2/3)*x + 1/3; f
1/17*x^19 - 2/3*x + 1/3
sage: f.denominator()
51

Finally, we try to compute the denominator of a polynomial with coefficients in the real numbers, which is a ring whose elements do not have a denominator method.

sage: R.<x> = RR[]
sage: f = x + RR('0.3'); f
x + 0.300000000000000
sage: f.denominator()
1.00000000000000

Check that the denominator is an element over the base whenever the base has no denominator function. This closes #9063.

sage: R.<a> = GF(5)[]
sage: x = R(0)
sage: x.denominator()
1
sage: type(x.denominator())
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
sage: isinstance(x.numerator() / x.denominator(), Polynomial)
True
sage: isinstance(x.numerator() / R(1), Polynomial)
False
derivative(*args)

The formal derivative of this polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: g = -x^4 + x^2/2 - x
sage: g.derivative()
-4*x^3 + x - 1
sage: g.derivative(x)
-4*x^3 + x - 1
sage: g.derivative(x, x)
-12*x^2 + 1
sage: g.derivative(x, 2)
-12*x^2 + 1
sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = PolynomialRing(R)
sage: f = t^3*x^2 + t^4*x^3
sage: f.derivative()
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(x)
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(t)
4*t^3*x^3 + 3*t^2*x^2
dict()

Return a sparse dictionary representation of this univariate polynomial.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 + -1/7*x + 13
sage: f.dict()
{0: 13, 1: -1/7, 3: 1}
diff(*args)

The formal derivative of this polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: g = -x^4 + x^2/2 - x
sage: g.derivative()
-4*x^3 + x - 1
sage: g.derivative(x)
-4*x^3 + x - 1
sage: g.derivative(x, x)
-12*x^2 + 1
sage: g.derivative(x, 2)
-12*x^2 + 1
sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = PolynomialRing(R)
sage: f = t^3*x^2 + t^4*x^3
sage: f.derivative()
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(x)
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(t)
4*t^3*x^3 + 3*t^2*x^2
differentiate(*args)

The formal derivative of this polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: g = -x^4 + x^2/2 - x
sage: g.derivative()
-4*x^3 + x - 1
sage: g.derivative(x)
-4*x^3 + x - 1
sage: g.derivative(x, x)
-12*x^2 + 1
sage: g.derivative(x, 2)
-12*x^2 + 1
sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = PolynomialRing(R)
sage: f = t^3*x^2 + t^4*x^3
sage: f.derivative()
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(x)
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(t)
4*t^3*x^3 + 3*t^2*x^2
discriminant()

Returns the discriminant of self.

The discriminant is

\[\begin{split}R_n := a_n^{2 n-2} \prod_{1<i<j<n} (r_i-r_j)^2,\end{split}\]

where \(n\) is the degree of self, \(a_n\) is the leading coefficient of self and the roots of self are \(r_1, \ldots, r_n\).

OUTPUT: An element of the base ring of the polynomial ring.

Note

Uses the identity \(R_n(f) := (-1)^{n (n-1)/2} R(f, f') a_n^{n-k-2}\), where \(n\) is the degree of self, \(a_n\) is the leading coefficient of self, \(f'\) is the derivative of \(f\), and \(k\) is the degree of \(f'\). Calls resultant().

EXAMPLES:

In the case of elliptic curves in special form, the discriminant is easy to calculate:

sage: R.<x> = QQ[]
sage: f = x^3 + x + 1
sage: d = f.discriminant(); d
-31
sage: d.parent() is QQ
True
sage: EllipticCurve([1, 1]).discriminant()/16
-31
sage: R.<x> = QQ[]
sage: f = 2*x^3 + x + 1
sage: d = f.discriminant(); d
-116

We can also compute discriminants over univariate and multivariate polynomial rings, provided that PARI’s variable ordering requirements are respected. Usually, your discriminants will work if you always ask for them in the variable x:

sage: R.<a> = QQ[]
sage: S.<x> = R[]
sage: f = a*x + x + a + 1
sage: d = f.discriminant(); d
1
sage: d.parent() is R
True
sage: R.<a, b> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a + b
sage: d = f.discriminant(); d
-4*a - 4*b
sage: d.parent() is R
True

Unfortunately Sage does not handle PARI’s variable ordering requirements gracefully, so the following has to be done through Singular:

sage: R.<x, y> = QQ[]
sage: S.<a> = R[]
sage: f = x^2 + a
sage: f.discriminant()
1

Check that trac ticket #13672 is fixed:

sage: R.<t> = GF(5)[]
sage: S.<x> = R[]
sage: f = x^10 + 2*x^6 + 2*x^5 + x + 2
sage: (f-t).discriminant()
4*t^5

The following examples show that trac ticket #11782 has been fixed:

sage: ZZ.quo(81)[x](3*x^2 + 3*x + 3).discriminant()
54
sage: ZZ.quo(9)[x](2*x^3 + x^2 + x).discriminant()
2

This was fixed by trac ticket #15422:

sage: R.<s> = PolynomialRing(Qp(2))
sage: (s^2).discriminant()
0

TESTS:

This was fixed by trac ticket #16014:

sage: PR.<b,t1,t2,x1,y1,x2,y2> = QQ[]
sage: PRmu.<mu> = PR[]
sage: E1 = diagonal_matrix(PR, [1, b^2, -b^2])
sage: M = matrix(PR, [[1,-t1,x1-t1*y1],[t1,1,y1+t1*x1],[0,0,1]])
sage: E1 = M.transpose()*E1*M
sage: E2 = E1.subs(t1=t2, x1=x2, y1=y2)
sage: det(mu*E1 + E2).discriminant().degrees()
(24, 12, 12, 8, 8, 8, 8)

This addresses an issue raised by trac ticket #15061:

sage: R.<T> = PowerSeriesRing(QQ)
sage: F = R([1,1],2)
sage: RP.<x> = PolynomialRing(R)
sage: P = x^2 - F
sage: P.discriminant()
4 + 4*T + O(T^2)
euclidean_degree()

Return the degree of this element as an element of a euclidean domain.

If this polynomial is defined over a field, this is simply its degree().

EXAMPLES:

sage: R.<x> = QQ[]
sage: x.euclidean_degree()
1
sage: R.<x> = ZZ[]
sage: x.euclidean_degree()
Traceback (most recent call last):
...
NotImplementedError
exponents()

Return the exponents of the monomials appearing in self.

EXAMPLES:

sage: _.<x> = PolynomialRing(ZZ)
sage: f = x^4+2*x^2+1
sage: f.exponents()
[0, 2, 4]
factor(**kwargs)

Return the factorization of self over its base ring.

INPUT:

  • kwargs – any keyword arguments are passed to the method _factor_univariate_polynomial() of the base ring if it defines such a method.

OUTPUT:

  • A factorization of self over its parent into a unit and irreducible factors. If the parent is a polynomial ring over a field, these factors are monic.

EXAMPLES:

Factorization is implemented over various rings. Over \(\QQ\):

sage: x = QQ['x'].0
sage: f = (x^3 - 1)^2
sage: f.factor()
(x - 1)^2 * (x^2 + x + 1)^2

Since \(\QQ\) is a field, the irreducible factors are monic:

sage: f = 10*x^5 - 1
sage: f.factor()
(10) * (x^5 - 1/10)
sage: f = 10*x^5 - 10
sage: f.factor()
(10) * (x - 1) * (x^4 + x^3 + x^2 + x + 1)

Over \(\ZZ\) the irreducible factors need not be monic:

sage: x = ZZ['x'].0
sage: f = 10*x^5 - 1
sage: f.factor()
10*x^5 - 1

We factor a non-monic polynomial over a finite field of 25 elements:

sage: k.<a> = GF(25)
sage: R.<x> = k[]
sage: f = 2*x^10 + 2*x + 2*a
sage: F = f.factor(); F
(2) * (x + a + 2) * (x^2 + 3*x + 4*a + 4) * (x^2 + (a + 1)*x + a + 2) * (x^5 + (3*a + 4)*x^4 + (3*a + 3)*x^3 + 2*a*x^2 + (3*a + 1)*x + 3*a + 1)

Notice that the unit factor is included when we multiply \(F\) back out:

sage: expand(F)
2*x^10 + 2*x + 2*a

A new ring. In the example below, we set the special method _factor_univariate_polynomial() in the base ring which is called to factor univariate polynomials. This facility can be used to easily extend polynomial factorization to work over new rings you introduce:

sage: R.<x> = PolynomialRing(IntegerModRing(4),implementation="NTL")
sage: (x^2).factor()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings with composite characteristic is not implemented
sage: R.base_ring()._factor_univariate_polynomial = lambda f: f.change_ring(ZZ).factor()
sage: (x^2).factor()
x^2
sage: del R.base_ring()._factor_univariate_polynomial # clean up

Arbitrary precision real and complex factorization:

sage: R.<x> = RealField(100)[]
sage: F = factor(x^2-3); F
(x - 1.7320508075688772935274463415) * (x + 1.7320508075688772935274463415)
sage: expand(F)
x^2 - 3.0000000000000000000000000000
sage: factor(x^2 + 1)
x^2 + 1.0000000000000000000000000000

sage: R.<x> = ComplexField(100)[]
sage: F = factor(x^2+3); F
(x - 1.7320508075688772935274463415*I) * (x + 1.7320508075688772935274463415*I)
sage: expand(F)
x^2 + 3.0000000000000000000000000000
sage: factor(x^2+1)
(x - I) * (x + I)
sage: f = R(I) * (x^2 + 1) ; f
I*x^2 + I
sage: F = factor(f); F
(1.0000000000000000000000000000*I) * (x - I) * (x + I)
sage: expand(F)
I*x^2 + I

Over a number field:

sage: K.<z> = CyclotomicField(15)
sage: x = polygen(K)
sage: ((x^3 + z*x + 1)^3*(x - z)).factor()
(x - z) * (x^3 + z*x + 1)^3
sage: cyclotomic_polynomial(12).change_ring(K).factor()
(x^2 - z^5 - 1) * (x^2 + z^5)
sage: ((x^3 + z*x + 1)^3*(x/(z+2) - 1/3)).factor()
(-1/331*z^7 + 3/331*z^6 - 6/331*z^5 + 11/331*z^4 - 21/331*z^3 + 41/331*z^2 - 82/331*z + 165/331) * (x - 1/3*z - 2/3) * (x^3 + z*x + 1)^3

Over a relative number field:

sage: x = polygen(QQ)
sage: K.<z> = CyclotomicField(3)
sage: L.<a> = K.extension(x^3 - 2)
sage: t = polygen(L, 't')
sage: f = (t^3 + t + a)*(t^5 + t + z); f
t^8 + t^6 + a*t^5 + t^4 + z*t^3 + t^2 + (a + z)*t + z*a
sage: f.factor()
(t^3 + t + a) * (t^5 + t + z)

Over the real double field:

sage: R.<x> = RDF[]
sage: (-2*x^2 - 1).factor()
(-2.0) * (x^2 + 0.5)
sage: (-2*x^2 - 1).factor().expand()
-2.0*x^2 - 1.0
sage: f = (x - 1)^3
sage: f.factor() # random output (unfortunately)
(x - 0.999990138083) * (x^2 - 2.00000986192*x + 1.00000986201)

The above output is incorrect because it relies on the roots() method, which does not detect that all the roots are real:

sage: f.roots() # random output (unfortunately)
[(0.999990138083, 1)]

Over the complex double field the factors are approximate and therefore occur with multiplicity 1:

sage: R.<x> = CDF[]
sage: f = (x^2 + 2*R(I))^3
sage: F = f.factor()
sage: F # random lower order digits
(x - 1.00000643627 + 1.00000715002*I) * (x - 1.00000297398 + 0.999990851041*I) * (x - 0.999990589745 + 1.00000199894*I) * (x + 0.999991986211 - 1.00000857983*I) * (x + 0.999996576554 - 0.999988769976*I) * (x + 1.00001143724 - 1.0000026502*I)
sage: [f(t[0][0]).abs() for t in F] # abs tol 1e-9
[1.979365054e-14, 1.97936298566e-14, 1.97936990747e-14, 3.6812407475e-14, 3.65211563729e-14, 3.65220890052e-14]

Factoring polynomials over \(\ZZ/n\ZZ\) for composite \(n\) is not implemented:

sage: R.<x> = PolynomialRing(Integers(35))
sage: f = (x^2+2*x+2)*(x^2+3*x+9)
sage: f.factor()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings with composite characteristic is not implemented

Factoring polynomials over the algebraic numbers (see trac ticket #8544):

sage: R.<x> = QQbar[]
sage: (x^8-1).factor()
(x - 1) * (x - 0.7071067811865475? - 0.7071067811865475?*I) * (x - 0.7071067811865475? + 0.7071067811865475?*I) * (x - I) * (x + I) * (x + 0.7071067811865475? - 0.7071067811865475?*I) * (x + 0.7071067811865475? + 0.7071067811865475?*I) * (x + 1)

Factoring polynomials over the algebraic reals (see trac ticket #8544):

sage: R.<x> = AA[]
sage: (x^8+1).factor()
(x^2 - 1.847759065022574?*x + 1.000000000000000?) * (x^2 - 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 1.847759065022574?*x + 1.000000000000000?)

TESTS:

This came up in ticket #7088:

sage: R.<x>=PolynomialRing(ZZ)
sage: f = 12*x^10 + x^9 + 432*x^3 + 9011
sage: g = 13*x^11 + 89*x^3 + 1
sage: F = f^2 * g^3
sage: F = f^2 * g^3; F.factor()
(12*x^10 + x^9 + 432*x^3 + 9011)^2 * (13*x^11 + 89*x^3 + 1)^3
sage: F = f^2 * g^3 * 7; F.factor()
7 * (12*x^10 + x^9 + 432*x^3 + 9011)^2 * (13*x^11 + 89*x^3 + 1)^3

This example came up in ticket #7097:

sage: x = polygen(QQ)
sage: f = 8*x^9 + 42*x^6 + 6*x^3 - 1
sage: g = x^24 - 12*x^23 + 72*x^22 - 286*x^21 + 849*x^20 - 2022*x^19 + 4034*x^18 - 6894*x^17 + 10182*x^16 - 13048*x^15 + 14532*x^14 - 13974*x^13 + 11365*x^12 - 7578*x^11 + 4038*x^10 - 1766*x^9 + 762*x^8 - 408*x^7 + 236*x^6 - 126*x^5 + 69*x^4 - 38*x^3 + 18*x^2 - 6*x + 1
sage: assert g.is_irreducible()
sage: K.<a> = NumberField(g)
sage: len(f.roots(K))
9
sage: f.factor()
(8) * (x^3 + 1/4) * (x^6 + 5*x^3 - 1/2)
sage: f.change_ring(K).factor()
(8) * (x - 3260097/3158212*a^22 + 35861067/3158212*a^21 - 197810817/3158212*a^20 + 722970825/3158212*a^19 - 1980508347/3158212*a^18 + 4374189477/3158212*a^17 - 4059860553/1579106*a^16 + 6442403031/1579106*a^15 - 17542341771/3158212*a^14 + 20537782665/3158212*a^13 - 20658463789/3158212*a^12 + 17502836649/3158212*a^11 - 11908953451/3158212*a^10 + 6086953981/3158212*a^9 - 559822335/789553*a^8 + 194545353/789553*a^7 - 505969453/3158212*a^6 + 338959407/3158212*a^5 - 155204647/3158212*a^4 + 79628015/3158212*a^3 - 57339525/3158212*a^2 + 26692783/3158212*a - 1636338/789553) * ...
sage: f = QQbar['x'](1)
sage: f.factor()
1

Factorization also works even if the variable of the finite field is nefariously labeled “x”:

sage: R.<x> = GF(3^2, 'x')[]
sage: f = x^10 +7*x -13
sage: G = f.factor(); G
(x + x) * (x + 2*x + 1) * (x^4 + (x + 2)*x^3 + (2*x + 2)*x + 2) * (x^4 + 2*x*x^3 + (x + 1)*x + 2)
sage: prod(G) == f
True
sage: R.<x0> = GF(9,'x')[]  # purposely calling it x to test robustness
sage: f = x0^3 + x0 + 1
sage: f.factor()
(x0 + 2) * (x0 + x) * (x0 + 2*x + 1)
sage: f = 0*x0
sage: f.factor()
Traceback (most recent call last):
...
ValueError: factorization of 0 not defined
sage: f = x0^0
sage: f.factor()
1

Over a complicated number field:

sage: x = polygen(QQ, 'x')
sage: f = x^6 + 10/7*x^5 - 867/49*x^4 - 76/245*x^3 + 3148/35*x^2 - 25944/245*x + 48771/1225
sage: K.<a> = NumberField(f)
sage: S.<T> = K[]
sage: ff = S(f); ff
T^6 + 10/7*T^5 - 867/49*T^4 - 76/245*T^3 + 3148/35*T^2 - 25944/245*T + 48771/1225
sage: F = ff.factor()
sage: len(F)
4
sage: F[:2]
[(T - a, 1), (T - 40085763200/924556084127*a^5 - 145475769880/924556084127*a^4 + 527617096480/924556084127*a^3 + 1289745809920/924556084127*a^2 - 3227142391585/924556084127*a - 401502691578/924556084127, 1)]
sage: expand(F)
T^6 + 10/7*T^5 - 867/49*T^4 - 76/245*T^3 + 3148/35*T^2 - 25944/245*T + 48771/1225
sage: f = x^2 - 1/3
sage: K.<a> = NumberField(f)
sage: A.<T> = K[]
sage: A(x^2 - 1).factor()
(T - 1) * (T + 1)
sage: A(3*x^2 - 1).factor()
(3) * (T - a) * (T + a)
sage: A(x^2 - 1/3).factor()
(T - a) * (T + a)

Test that ticket #10279 is fixed:

sage: R.<t> = PolynomialRing(QQ)
sage: K.<a> = NumberField(t^4 - t^2 + 1)
sage: pol = t^3 + (-4*a^3 + 2*a)*t^2 - 11/3*a^2*t + 2/3*a^3 - 4/3*a
sage: pol.factor()
(t - 2*a^3 + a) * (t - 4/3*a^3 + 2/3*a) * (t - 2/3*a^3 + 1/3*a)

Test that this factorization really uses nffactor() internally:

sage: pari.default("debug", 3)
sage: F = pol.factor()

Entering nffactor:
...
sage: pari.default("debug", 0)

Test that ticket #10369 is fixed:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
sage: R.<t> = PolynomialRing(K)

sage: pol = (-1/7*a^5 - 1/7*a^4 - 1/7*a^3 - 1/7*a^2 - 2/7*a - 1/7)*t^10 + (4/7*a^5 - 2/7*a^4 - 2/7*a^3 - 2/7*a^2 - 2/7*a - 6/7)*t^9 + (90/49*a^5 + 152/49*a^4 + 18/49*a^3 + 24/49*a^2 + 30/49*a + 36/49)*t^8 + (-10/49*a^5 + 10/7*a^4 + 198/49*a^3 - 102/49*a^2 - 60/49*a - 26/49)*t^7 + (40/49*a^5 + 45/49*a^4 + 60/49*a^3 + 277/49*a^2 - 204/49*a - 78/49)*t^6 + (90/49*a^5 + 110/49*a^4 + 2*a^3 + 80/49*a^2 + 46/7*a - 30/7)*t^5 + (30/7*a^5 + 260/49*a^4 + 250/49*a^3 + 232/49*a^2 + 32/7*a + 8)*t^4 + (-184/49*a^5 - 58/49*a^4 - 52/49*a^3 - 66/49*a^2 - 72/49*a - 72/49)*t^3 + (18/49*a^5 - 32/49*a^4 + 10/49*a^3 + 4/49*a^2)*t^2 + (2/49*a^4 - 4/49*a^3 + 2/49*a^2)*t
sage: pol.factor()
(-1/7*a^5 - 1/7*a^4 - 1/7*a^3 - 1/7*a^2 - 2/7*a - 1/7) * t * (t - a^5 - a^4 - a^3 - a^2 - a - 1)^4 * (t^5 + (-12/7*a^5 - 10/7*a^4 - 8/7*a^3 - 6/7*a^2 - 4/7*a - 2/7)*t^4 + (12/7*a^5 - 8/7*a^3 + 16/7*a^2 + 2/7*a + 20/7)*t^3 + (-20/7*a^5 - 20/7*a^3 - 20/7*a^2 + 4/7*a - 2)*t^2 + (12/7*a^5 + 12/7*a^3 + 2/7*a + 16/7)*t - 4/7*a^5 - 4/7*a^3 - 4/7*a - 2/7)

sage: pol = (1/7*a^2 - 1/7*a)*t^10 + (4/7*a - 6/7)*t^9 + (102/49*a^5 + 99/49*a^4 + 96/49*a^3 + 93/49*a^2 + 90/49*a + 150/49)*t^8 + (-160/49*a^5 - 36/49*a^4 - 48/49*a^3 - 8/7*a^2 - 60/49*a - 60/49)*t^7 + (30/49*a^5 - 55/49*a^4 + 20/49*a^3 + 5/49*a^2)*t^6 + (6/49*a^4 - 12/49*a^3 + 6/49*a^2)*t^5
sage: pol.factor()
(1/7*a^2 - 1/7*a) * t^5 * (t^5 + (-40/7*a^5 - 38/7*a^4 - 36/7*a^3 - 34/7*a^2 - 32/7*a - 30/7)*t^4 + (60/7*a^5 - 30/7*a^4 - 18/7*a^3 - 9/7*a^2 - 3/7*a)*t^3 + (60/7*a^4 - 40/7*a^3 - 16/7*a^2 - 4/7*a)*t^2 + (30/7*a^3 - 25/7*a^2 - 5/7*a)*t + 6/7*a^2 - 6/7*a)

sage: pol = x^10 + (4/7*a - 6/7)*x^9 + (9/49*a^2 - 3/7*a + 15/49)*x^8 + (8/343*a^3 - 32/343*a^2 + 40/343*a - 20/343)*x^7 + (5/2401*a^4 - 20/2401*a^3 + 40/2401*a^2 - 5/343*a + 15/2401)*x^6 + (-6/16807*a^4 + 12/16807*a^3 - 18/16807*a^2 + 12/16807*a - 6/16807)*x^5
sage: pol.factor()
x^5 * (x^5 + (4/7*a - 6/7)*x^4 + (9/49*a^2 - 3/7*a + 15/49)*x^3 + (8/343*a^3 - 32/343*a^2 + 40/343*a - 20/343)*x^2 + (5/2401*a^4 - 20/2401*a^3 + 40/2401*a^2 - 5/343*a + 15/2401)*x - 6/16807*a^4 + 12/16807*a^3 - 18/16807*a^2 + 12/16807*a - 6/16807)

Factoring over a number field over which we cannot factor the discriminant by trial division:

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^16 - x - 6)
sage: R.<x> = PolynomialRing(K)
sage: f = (x+a)^50 - (a-1)^50
sage: len(factor(f))
6
sage: pari(K.discriminant()).factor(limit=0)
[-1, 1; 3, 15; 23, 1; 887, 1; 12583, 1; 2354691439917211, 1]
sage: factor(K.discriminant())
-1 * 3^15 * 23 * 887 * 12583 * 6335047 * 371692813

Factoring over a number field over which we cannot factor the discriminant and over which \(nffactor()\) fails:

sage: p = next_prime(10^50); q = next_prime(10^51); n = p*q;
sage: K.<a> = QuadraticField(p*q)
sage: R.<x> = PolynomialRing(K)
sage: K.pari_polynomial('a').nffactor("x^2+1")
Traceback (most recent call last):
...
PariError: precision too low in floorr (precision loss in truncation)
sage: factor(x^2 + 1)
x^2 + 1
sage: factor( (x - a) * (x + 2*a) )
(x - a) * (x + 2*a)

A test where nffactor used to fail without a nf structure:

sage: x = polygen(QQ)
sage: K = NumberField([x^2-1099511627777, x^3-3],'a')
sage: x = polygen(K)
sage: f = x^3 - 3
sage: factor(f)
(x - a1) * (x^2 + a1*x + a1^2)

We check that trac ticket #7554 is fixed:

sage: L.<q> = LaurentPolynomialRing(QQ)
sage: F = L.fraction_field()
sage: R.<x> = PolynomialRing(F)
sage: factor(x)
x
sage: factor(x^2 - q^2)
(-1) * (-x + q) * (x + q)
sage: factor(x^2 - q^-2)
(1/q^2) * (q*x - 1) * (q*x + 1)

sage: P.<a,b,c> = PolynomialRing(ZZ)
sage: R.<x> = PolynomialRing(FractionField(P))
sage: p = (x - a)*(b*x + c)*(a*b*x + a*c) / (a + 2)
sage: factor(p)
(a/(a + 2)) * (x - a) * (b*x + c)^2
hamming_weight()

Returns the number of non-zero coefficients of self.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: f = x^3 - x
sage: f.hamming_weight()
2
sage: R(0).hamming_weight()
0
sage: f = (x+1)^100
sage: f.hamming_weight()
101
sage: S = GF(5)['y']
sage: S(f).hamming_weight()
5
sage: cyclotomic_polynomial(105).hamming_weight()
33
homogenize(var='h')

Return the homogenization of this polynomial.

The polynomial itself is returned if it homogeneous already. Otherwise, its monomials are multiplied with the smallest powers of var such that they all have the same total degree.

INPUT:

  • var – a variable in the polynomial ring (as a string, an element of the ring, or 0) or a name for a new variable (default: 'h')

OUTPUT:

If var specifies the variable in the polynomial ring, then a homogeneous element in that ring is returned. Otherwise, a homogeneous element is returned in a polynomial ring with an extra last variable var.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^2 + 1
sage: f.homogenize()
x^2 + h^2

The parameter var can be used to specify the name of the variable:

sage: g = f.homogenize('z'); g
x^2 + z^2
sage: g.parent()
Multivariate Polynomial Ring in x, z over Rational Field

However, if the polynomial is homogeneous already, then that parameter is ignored and no extra variable is added to the polynomial ring:

sage: f = x^2
sage: g = f.homogenize('z'); g
x^2
sage: g.parent()
Univariate Polynomial Ring in x over Rational Field

For compatibility with the multivariate case, if var specifies the variable of the polynomial ring, then the monomials are multiplied with the smallest powers of var such that the result is homogeneous; in other words, we end up with a monomial whose leading coefficient is the sum of the coefficients of the polynomial:

sage: f = x^2 + x + 1
sage: f.homogenize('x')
3*x^2

In positive characterstic, the degree can drop in this case:

sage: R.<x> = GF(2)[]
sage: f = x + 1
sage: f.homogenize(x)
0

For compatibility with the multivariate case, the parameter var can also be 0 to specify the variable in the polynomial ring:

sage: R.<x> = QQ[]
sage: f = x^2 + x + 1
sage: f.homogenize(0)
3*x^2
integral(var=None)

Return the integral of this polynomial.

By default, the integration variable is the variable of the polynomial.

Otherwise, the integration variable is the optional parameter var

Note

The integral is always chosen so the constant term is 0.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R(0).integral()
0
sage: f = R(2).integral(); f
2*x

Note that the integral lives over the fraction field of the scalar coefficients:

sage: f.parent()
Univariate Polynomial Ring in x over Rational Field
sage: R(0).integral().parent()
Univariate Polynomial Ring in x over Rational Field

sage: f = x^3 + x - 2
sage: g = f.integral(); g
1/4*x^4 + 1/2*x^2 - 2*x
sage: g.parent()
Univariate Polynomial Ring in x over Rational Field

This shows that the issue at trac ticket #7711 is resolved:

sage: P.<x,z> = PolynomialRing(GF(2147483647))
sage: Q.<y> = PolynomialRing(P)
sage: p=x+y+z
sage: p.integral()
-1073741823*y^2 + (x + z)*y

sage: P.<x,z> = PolynomialRing(GF(next_prime(2147483647)))
sage: Q.<y> = PolynomialRing(P)
sage: p=x+y+z
sage: p.integral()
1073741830*y^2 + (x + z)*y

A truly convoluted example:

sage: A.<a1, a2> = PolynomialRing(ZZ)
sage: B.<b> = PolynomialRing(A)
sage: C.<c> = PowerSeriesRing(B)
sage: R.<x> = PolynomialRing(C)
sage: f = a2*x^2 + c*x - a1*b
sage: f.parent()
Univariate Polynomial Ring in x over Power Series Ring in c
over Univariate Polynomial Ring in b over Multivariate Polynomial
Ring in a1, a2 over Integer Ring
sage: f.integral()
1/3*a2*x^3 + 1/2*c*x^2 - a1*b*x
sage: f.integral().parent()
Univariate Polynomial Ring in x over Power Series Ring in c
over Univariate Polynomial Ring in b over Multivariate Polynomial
Ring in a1, a2 over Rational Field
sage: g = 3*a2*x^2 + 2*c*x - a1*b
sage: g.integral()
a2*x^3 + c*x^2 - a1*b*x
sage: g.integral().parent()
Univariate Polynomial Ring in x over Power Series Ring in c
over Univariate Polynomial Ring in b over Multivariate Polynomial
Ring in a1, a2 over Rational Field

Integration with respect to a variable in the base ring:

sage: R.<x> = QQ[]
sage: t = PolynomialRing(R,'t').gen()
sage: f = x*t +5*t^2
sage: f.integral(x)
5*x*t^2 + 1/2*x^2*t
inverse_mod(a, m)

Inverts the polynomial a with respect to m, or raises a ValueError if no such inverse exists. The parameter m may be either a single polynomial or an ideal (for consistency with inverse_mod in other rings).

EXAMPLES:

sage: S.<t> = QQ[]
sage: f = inverse_mod(t^2 + 1, t^3 + 1); f
-1/2*t^2 - 1/2*t + 1/2
sage: f * (t^2 + 1) % (t^3 + 1)
1
sage: f = t.inverse_mod((t+1)^7); f
-t^6 - 7*t^5 - 21*t^4 - 35*t^3 - 35*t^2 - 21*t - 7
sage: (f * t) + (t+1)^7
1
sage: t.inverse_mod(S.ideal((t + 1)^7)) == f
True

This also works over inexact rings, but note that due to rounding error the product may not always exactly equal the constant polynomial 1 and have extra terms with coefficients close to zero.

sage: R.<x> = RDF[]
sage: epsilon = RDF(1).ulp()*50   # Allow an error of up to 50 ulp
sage: f = inverse_mod(x^2 + 1, x^5 + x + 1); f
0.4*x^4 - 0.2*x^3 - 0.4*x^2 + 0.2*x + 0.8
sage: poly = f * (x^2 + 1) % (x^5 + x + 1)
sage: # Remove noisy zero terms:
sage: parent(poly)([ 0.0 if abs(c)<=epsilon else c for c in poly.coeffs() ])
1.0
sage: f = inverse_mod(x^3 - x + 1, x - 2); f
0.142857142857
sage: f * (x^3 - x + 1) % (x - 2)
1.0
sage: g = 5*x^3+x-7; m = x^4-12*x+13; f = inverse_mod(g, m); f
-0.0319636125...*x^3 - 0.0383269759...*x^2 - 0.0463050900...*x + 0.346479687...
sage: poly = f*g % m
sage: # Remove noisy zero terms:
sage: parent(poly)([ 0.0 if abs(c)<=epsilon else c for c in poly.coeffs() ])
1.0

ALGORITHM: Solve the system as + mt = 1, returning s as the inverse of a mod m.

Uses the Euclidean algorithm for exact rings, and solves a linear system for the coefficients of s and t for inexact rings (as the Euclidean algorithm may not converge in that case).

AUTHORS:

  • Robert Bradshaw (2007-05-31)
inverse_of_unit()

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x - 90283
sage: f.inverse_of_unit()
Traceback (most recent call last):
...
ValueError: self is not a unit.
sage: f = R(-90283); g = f.inverse_of_unit(); g
-1/90283
sage: parent(g)
Univariate Polynomial Ring in x over Rational Field
is_constant()

Return True if this is a constant polynomial.

OUTPUT:

  • bool - True if and only if this polynomial is constant

EXAMPLES:

sage: R.<x> = ZZ[]
sage: x.is_constant()
False
sage: R(2).is_constant()
True
sage: R(0).is_constant()
True
is_cyclotomic()

Return True if self is a cyclotomic polynomial.

A cyclotomic polynomial is a monic, irreducible polynomial such that all roots are roots of unity.

ALGORITHM:

The first cyclotomic polynomial x-1 is treated apart, otherwise the first algorithm of [BD89] is used.

EXAMPLES:

Quick tests:

sage: P.<x> = ZZ[x]
sage: (x - 1).is_cyclotomic()
True
sage: (x + 1).is_cyclotomic()
True
sage: (x^2 - 1).is_cyclotomic()
False

Test first 100 cyclotomic polynomials:

sage: all(cyclotomic_polynomial(i).is_cyclotomic() for i in xrange(1,101))
True

Some more tests:

sage: (x^16 + x^14 - x^10 + x^8 - x^6 + x^2 + 1).is_cyclotomic()
False
sage: (x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1).is_cyclotomic()
True
sage: y = polygen(QQ)
sage: (y/2 - 1/2).is_cyclotomic()
False
sage: (2*(y/2 - 1/2)).is_cyclotomic()
True

Test using other rings:

sage: z = polygen(GF(5))
sage: (z - 1).is_cyclotomic()
Traceback (most recent call last):
...
NotImplementedError: not implemented in non-zero characteristic

REFERENCES:

[BD89]R. J. Bradford and J. H. Davenport, Effective tests for cyclotomic polynomials, Symbolic and Algebraic Computation (1989) pp. 244 – 251, doi:10.1007/3-540-51084-2_22
is_gen()

Return True if this polynomial is the distinguished generator of the parent polynomial ring.

EXAMPLES:

sage: R.<x> = QQ[]
sage: R(1).is_gen()
False
sage: R(x).is_gen()
True

Important - this function doesn’t return True if self equals the generator; it returns True if self is the generator.

sage: f = R([0,1]); f
x
sage: f.is_gen()
False
sage: f is x
False
sage: f == x
True
is_homogeneous()

Return True if this polynomial is homogeneous.

EXAMPLES:

sage: P.<x> = PolynomialRing(QQ)
sage: x.is_homogeneous()
True
sage: P(0).is_homogeneous()
True
sage: (x+1).is_homogeneous()
False
is_irreducible()

Return True precisely if this polynomial is irreducible over its base ring.

Testing irreducibility over \(\ZZ/n\ZZ\) for composite \(n\) is not implemented.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: (x^3 + 1).is_irreducible()
False
sage: (x^2 - 1).is_irreducible()
False
sage: (x^3 + 2).is_irreducible()
True
sage: R(0).is_irreducible()
False

See trac ticket #5140,

sage: R(1).is_irreducible()
False
sage: R(4).is_irreducible()
False
sage: R(5).is_irreducible()
True

The base ring does matter: for example, 2x is irreducible as a polynomial in QQ[x], but not in ZZ[x],

sage: R.<x> = ZZ[]
sage: R(2*x).is_irreducible()
False
sage: R.<x> = QQ[]
sage: R(2*x).is_irreducible()
True

TESTS:

sage: F.<t> = NumberField(x^2-5)
sage: Fx.<xF> = PolynomialRing(F)
sage: f = Fx([2*t - 5, 5*t - 10, 3*t - 6, -t, -t + 2, 1])
sage: f.is_irreducible()
False
sage: f = Fx([2*t - 3, 5*t - 10, 3*t - 6, -t, -t + 2, 1])
sage: f.is_irreducible()
True
is_monic()

Returns True if this polynomial is monic. The zero polynomial is by definition not monic.

EXAMPLES:

sage: x = QQ['x'].0
sage: f = x + 33
sage: f.is_monic()
True
sage: f = 0*x
sage: f.is_monic()
False
sage: f = 3*x^3 + x^4 + x^2
sage: f.is_monic()
True
sage: f = 2*x^2 + x^3 + 56*x^5
sage: f.is_monic()
False

AUTHORS:

  • Naqi Jaffery (2006-01-24): examples
is_monomial()

Returns True if self is a monomial, i.e., a power of the generator.

EXAMPLES:

sage: R.<x> = QQ[]
sage: x.is_monomial()
True
sage: (x+1).is_monomial()
False
sage: (x^2).is_monomial()
True
sage: R(1).is_monomial()
True

The coefficient must be 1:

sage: (2*x^5).is_monomial()
False

To allow a non-1 leading coefficient, use is_term():

sage: (2*x^5).is_term()
True

Warning

The definition of is_monomial in Sage up to 4.7.1 was the same as is_term, i.e., it allowed a coefficient not equal to 1.

is_nilpotent()

Return True if this polynomial is nilpotent.

EXAMPLES:

sage: R = Integers(12)
sage: S.<x> = R[]
sage: f = 5 + 6*x
sage: f.is_nilpotent()
False
sage: f = 6 + 6*x^2
sage: f.is_nilpotent()
True
sage: f^2
0

EXERCISE (Atiyah-McDonald, Ch 1): Let \(A[x]\) be a polynomial ring in one variable. Then \(f=\sum a_i x^i \in A[x]\) is nilpotent if and only if every \(a_i\) is nilpotent.

is_primitive(n=None, n_prime_divs=None)

Returns True if the polynomial is primitive. The semantics of “primitive” depend on the polynomial coefficients.

  • (field theory) A polynomial of degree \(m\) over a finite field \(\GF{q}\) is primitive if it is irreducible and its root in \(\GF{q^m}\) generates the multiplicative group \(\GF{q^m}^*\).
  • (ring theory) A polynomial over a ring is primitive if its coefficients generate the unit ideal.

Calling \(is_primitive\) on a polynomial over an infinite field will raise an error.

The additional inputs to this function are to speed up computation for field semantics (see note).

INPUTS:

  • n (default: None) - if provided, should equal \(q-1\) where self.parent() is the field with \(q\) elements; otherwise it will be computed.
  • n_prime_divs (default: None) - if provided, should be a list of the prime divisors of n; otherwise it will be computed.

Note

Computation of the prime divisors of n can dominate the running time of this method, so performing this computation externally (e.g. pdivs=n.prime_divisors()) is a good idea for repeated calls to is_primitive for polynomials of the same degree.

Results may be incorrect if the wrong n and/or factorization are provided.

EXAMPLES:

Field semantics examples.

::

  sage: R.<x> = GF(2)['x']
  sage: f = x^4+x^3+x^2+x+1
  sage: f.is_irreducible(), f.is_primitive()
  (True, False)
  sage: f = x^3+x+1
  sage: f.is_irreducible(), f.is_primitive()
  (True, True)
  sage: R.<x> = GF(3)[]
  sage: f = x^3-x+1
  sage: f.is_irreducible(), f.is_primitive()
  (True, True)
  sage: f = x^2+1
  sage: f.is_irreducible(), f.is_primitive()
  (True, False)
  sage: R.<x> = GF(5)[]
  sage: f = x^2+x+1
  sage: f.is_primitive()
  False
  sage: f = x^2-x+2
  sage: f.is_primitive()
  True
  sage: x=polygen(QQ); f=x^2+1
  sage: f.is_primitive()
  Traceback (most recent call last):
  ...
  NotImplementedError: is_primitive() not defined for polynomials over infinite fields.

Ring semantics examples.

::

  sage: x=polygen(ZZ)
  sage: f = 5*x^2+2
  sage: f.is_primitive()
  True
  sage: f = 5*x^2+5
  sage: f.is_primitive()
  False

  sage: K=NumberField(x^2+5,'a')
  sage: R=K.ring_of_integers()
  sage: a=R.gen(1)
  sage: a^2
  -5
  sage: f=a*x+2
  sage: f.is_primitive()
  True
  sage: f=(1+a)*x+2
  sage: f.is_primitive()
  False

  sage: x=polygen(Integers(10));
  sage: f=5*x^2+2
  sage: #f.is_primitive()  #BUG:: elsewhere in Sage, should return True
  sage: f=4*x^2+2
  sage: #f.is_primitive()  #BUG:: elsewhere in Sage, should return False

TESTS:

sage: R.<x> = GF(2)['x']
sage: f = x^4+x^3+x^2+x+1
sage: f.is_primitive(15)
False
sage: f.is_primitive(15, [3,5])
False
sage: f.is_primitive(n_prime_divs=[3,5])
False
sage: f = x^3+x+1
sage: f.is_primitive(7, [7])
True
sage: R.<x> = GF(3)[]
sage: f = x^3-x+1
sage: f.is_primitive(26, [2,13])
True
sage: f = x^2+1
sage: f.is_primitive(8, [2])
False
sage: R.<x> = GF(5)[]
sage: f = x^2+x+1
sage: f.is_primitive(24, [2,3])
False
sage: f = x^2-x+2
sage: f.is_primitive(24, [2,3])
True
sage: x=polygen(Integers(103)); f=x^2+1
sage: f.is_primitive()
False
is_square(root=False)

Returns whether or not polynomial is square. If the optional argument root is True, then also returns the square root (or None, if the polynomial is not square).

INPUT:

  • root - whether or not to also return a square root (default: False)

OUTPUT:

  • bool - whether or not a square
  • object - (optional) an actual square root if found, and None otherwise.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: f = 12*(x+1)^2 * (x+3)^2
sage: S.<y> = PolynomialRing(RR)
sage: g = 12*(y+1)^2 * (y+3)^2
sage: f.is_square()
False
sage: f.is_square(root=True)
(False, None)
sage: g.is_square()
True
sage: h = f/3; h
4*x^4 + 32*x^3 + 88*x^2 + 96*x + 36
sage: h.is_square(root=True)
(True, 2*x^2 + 8*x + 6)

TESTS:

Make sure ticket #9093 is fixed:

sage: R(1).is_square()
True
sage: R(4/9).is_square()
True
sage: R(-1/3).is_square()
False
sage: R(0).is_square()
True
is_squarefree()

Return False if this polynomial is not square-free, i.e., if there is a non-unit \(g\) in the polynomial ring such that \(g^2\) divides self.

Warning

This method is not consistent with squarefree_decomposition() since the latter does not factor the content of a polynomial. See the examples below.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = (x-1)*(x-2)*(x^2-5)*(x^17-3); f
x^21 - 3*x^20 - 3*x^19 + 15*x^18 - 10*x^17 - 3*x^4 + 9*x^3 + 9*x^2 - 45*x + 30
sage: f.is_squarefree()
True
sage: (f*(x^2-5)).is_squarefree()
False

A generic implementation is available for polynomials defined over principal ideal domains of characteristic 0; the algorithm relies on gcd computations:

sage: R.<x> = ZZ[]
sage: (2*x).is_squarefree()
True
sage: (4*x).is_squarefree()
False
sage: (2*x^2).is_squarefree()
False
sage: R(0).is_squarefree()
False

sage: S.<y> = QQ[]
sage: R.<x> = S[]
sage: (2*x*y).is_squarefree() # R does not provide a gcd implementation
Traceback (most recent call last):
...
AttributeError: 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense' object has no attribute 'gcd'
sage: (2*x*y^2).is_squarefree()
False

Over principal ideal domains of positive characteristic, we compute the square-free decomposition or a full factorization depending on which is available:

sage: K.<t> = FunctionField(GF(3))
sage: R.<x> = K[]
sage: (x^3-x).is_squarefree()
True
sage: (x^3-1).is_squarefree()
False
sage: (x^3+t).is_squarefree()
True
sage: (x^3+t^3).is_squarefree()
False

In the following example, \(t^2\) is a unit in the base field:

sage: R(t^2).is_squarefree()
True

This method is not consistent with squarefree_decomposition():

sage: R.<x> = ZZ[]
sage: f = 4 * x
sage: f.is_squarefree()
False
sage: f.squarefree_decomposition()
(4) * x

If you want this method equally not to consider the content, you can remove it as in the following example:

sage: c = f.content()
sage: (f/c).is_squarefree()
True
is_term()

Return True if self is an element of the base ring times a power of the generator.

EXAMPLES:

sage: R.<x> = QQ[]
sage: x.is_term()
True
sage: R(1).is_term()
True
sage: (3*x^5).is_term()
True
sage: (1+3*x^5).is_term()
False

To require that the coefficient is 1, use is_monomial() instead:

sage: (3*x^5).is_monomial()
False
is_unit()

Return True if this polynomial is a unit.

EXAMPLES:

sage: a = Integers(90384098234^3)
sage: b = a(2*191*236607587)
sage: b.is_nilpotent()
True
sage: R.<x> = a[]
sage: f = 3 + b*x + b^2*x^2
sage: f.is_unit()
True
sage: f = 3 + b*x + b^2*x^2 + 17*x^3
sage: f.is_unit()
False

EXERCISE (Atiyah-McDonald, Ch 1): Let \(A[x]\) be a polynomial ring in one variable. Then \(f=\sum a_i x^i \in A[x]\) is a unit if and only if \(a_0\) is a unit and \(a_1,\ldots, a_n\) are nilpotent.

lcm(other)

Let f and g be two polynomials. Then this function returns the monic least common multiple of f and g.

leading_coefficient()

Return the leading coefficient of this polynomial.

OUTPUT: element of the base ring

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = (-2/5)*x^3 + 2*x - 1/3
sage: f.leading_coefficient()
-2/5
list()

Return a new copy of the list of the underlying elements of self.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = (-2/5)*x^3 + 2*x - 1/3
sage: v = f.list(); v
[-1/3, 2, 0, -2/5]

Note that v is a list, it is mutable, and each call to the list method returns a new list:

sage: type(v)
<type 'list'>
sage: v[0] = 5
sage: f.list()
[-1/3, 2, 0, -2/5]

Here is an example with a generic polynomial ring:

sage: R.<x> = QQ[]
sage: S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: v = f.list(); v
[-3*x, x, 0, 1]
sage: v[0] = 10
sage: f.list()
[-3*x, x, 0, 1]
map_coefficients(f, new_base_ring=None)

Returns the polynomial obtained by applying f to the non-zero coefficients of self.

If f is a sage.categories.map.Map, then the resulting polynomial will be defined over the codomain of f. Otherwise, the resulting polynomial will be over the same ring as self. Set new_base_ring to override this behaviour.

INPUT:

  • f – a callable that will be applied to the coefficients of self.
  • new_base_ring (optional) – if given, the resulting polynomial will be defined over this ring.

EXAMPLES:

sage: R.<x> = SR[]
sage: f = (1+I)*x^2 + 3*x - I
sage: f.map_coefficients(lambda z: z.conjugate())
(-I + 1)*x^2 + 3*x + I
sage: R.<x> = ZZ[]
sage: f = x^2 + 2
sage: f.map_coefficients(lambda a: a + 42)
43*x^2 + 44
sage: R.<x> = PolynomialRing(SR, sparse=True)
sage: f = (1+I)*x^(2^32) - I
sage: f.map_coefficients(lambda z: z.conjugate())
(-I + 1)*x^4294967296 + I
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: f = x^(2^32) + 2
sage: f.map_coefficients(lambda a: a + 42)
43*x^4294967296 + 44

Examples with different base ring:

sage: R.<x> = ZZ[]
sage: k = GF(2)
sage: residue = lambda x: k(x)
sage: f = 4*x^2+x+3
sage: g = f.map_coefficients(residue); g
x + 1
sage: g.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: g = f.map_coefficients(residue, new_base_ring = k); g
x + 1
sage: g.parent()
Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL)
sage: residue = k.coerce_map_from(ZZ)
sage: g = f.map_coefficients(residue); g
x + 1
sage: g.parent()
Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL)
mod(other)

Remainder of division of self by other.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: x % (x+1)
-1
sage: (x^3 + x - 1) % (x^2 - 1)
2*x - 1
monic()

Return this polynomial divided by its leading coefficient. Does not change this polynomial.

EXAMPLES:

sage: x = QQ['x'].0
sage: f = 2*x^2 + x^3 + 56*x^5
sage: f.monic()
x^5 + 1/56*x^3 + 1/28*x^2
sage: f = (1/4)*x^2 + 3*x + 1
sage: f.monic()
x^2 + 12*x + 4

The following happens because \(f = 0\) cannot be made into a monic polynomial

sage: f = 0*x
sage: f.monic()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero

Notice that the monic version of a polynomial over the integers is defined over the rationals.

sage: x = ZZ['x'].0
sage: f = 3*x^19 + x^2 - 37
sage: g = f.monic(); g
x^19 + 1/3*x^2 - 37/3
sage: g.parent()
Univariate Polynomial Ring in x over Rational Field

AUTHORS:

  • Naqi Jaffery (2006-01-24): examples
newton_raphson(n, x0)

Return a list of n iterative approximations to a root of this polynomial, computed using the Newton-Raphson method.

The Newton-Raphson method is an iterative root-finding algorithm. For f(x) a polynomial, as is the case here, this is essentially the same as Horner’s method.

INPUT:

  • n - an integer (=the number of iterations),
  • x0 - an initial guess x0.

OUTPUT: A list of numbers hopefully approximating a root of f(x)=0.

If one of the iterates is a critical point of f then a ZeroDivisionError exception is raised.

EXAMPLES:

sage: x = PolynomialRing(RealField(), 'x').gen()
sage: f = x^2 - 2
sage: f.newton_raphson(4, 1)
[1.50000000000000, 1.41666666666667, 1.41421568627451, 1.41421356237469]

AUTHORS:

  • David Joyner and William Stein (2005-11-28)
newton_slopes(p)

Return the \(p\)-adic slopes of the Newton polygon of self, when this makes sense.

OUTPUT: list of rational numbers

EXAMPLES:

sage: x = QQ['x'].0
sage: f = x^3 + 2
sage: f.newton_slopes(2)
[1/3, 1/3, 1/3]

ALGORITHM: Uses PARI.

norm(p)

Return the \(p\)-norm of this polynomial.

DEFINITION: For integer \(p\), the \(p\)-norm of a polynomial is the \(p\)th root of the sum of the \(p\)th powers of the absolute values of the coefficients of the polynomial.

INPUT:

  • p - (positive integer or +infinity) the degree of the norm

EXAMPLES:

sage: R.<x> = RR[]
sage: f = x^6 + x^2 + -x^4 - 2*x^3
sage: f.norm(2)
2.64575131106459
sage: (sqrt(1^2 + 1^2 + (-1)^2 + (-2)^2)).n()
2.64575131106459
sage: f.norm(1)
5.00000000000000
sage: f.norm(infinity)
2.00000000000000
sage: f.norm(-1)
Traceback (most recent call last):
...
ValueError: The degree of the norm must be positive

TESTS:

sage: R.<x> = RR[]
sage: f = x^6 + x^2 + -x^4 -x^3
sage: f.norm(int(2))
2.00000000000000

AUTHORS:

  • Didier Deshommes
  • William Stein: fix bugs, add definition, etc.
numerator()

Return a numerator of self computed as self * self.denominator()

Note that some subclases may implement its own numerator function. For example, see sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint

Warning

This is not the numerator of the rational function defined by self, which would always be self since self is a polynomial.

EXAMPLES:

First we compute the numerator of a polynomial with integer coefficients, which is of course self.

sage: R.<x> = ZZ[]
sage: f = x^3 + 17*x + 1
sage: f.numerator()
x^3 + 17*x + 1
sage: f == f.numerator()
True

Next we compute the numerator of a polynomial with rational coefficients.

sage: R.<x> = PolynomialRing(QQ)
sage: f = (1/17)*x^19 - (2/3)*x + 1/3; f
1/17*x^19 - 2/3*x + 1/3
sage: f.numerator()
3*x^19 - 34*x + 17
sage: f == f.numerator()
False

We try to compute the denominator of a polynomial with coefficients in the real numbers, which is a ring whose elements do not have a denominator method.

sage: R.<x> = RR[]
sage: f = x + RR('0.3'); f
x + 0.300000000000000
sage: f.numerator()
x + 0.300000000000000

We check that the computation the numerator and denominator are valid

sage: K=NumberField(symbolic_expression('x^3+2'),'a')['s,t']['x']
sage: f=K.random_element()
sage: f.numerator() / f.denominator() == f
True
sage: R=RR['x']
sage: f=R.random_element()
sage: f.numerator() / f.denominator() == f
True
ord(p=None)

This is the same as the valuation of self at p. See the documentation for self.valuation.

EXAMPLES:

sage: P,x=PolynomialRing(ZZ,'x').objgen()
sage: (x^2+x).ord(x+1)
1
padded_list(n=None)

Return list of coefficients of self up to (but not including) \(q^n\).

Includes 0’s in the list on the right so that the list has length \(n\).

INPUT:

  • n - (default: None); if given, an integer that is at least 0

EXAMPLES:

sage: x = polygen(QQ)
sage: f = 1 + x^3 + 23*x^5
sage: f.padded_list()
[1, 0, 0, 1, 0, 23]
sage: f.padded_list(10)
[1, 0, 0, 1, 0, 23, 0, 0, 0, 0]
sage: len(f.padded_list(10))
10
sage: f.padded_list(3)
[1, 0, 0]
sage: f.padded_list(0)
[]
sage: f.padded_list(-1)
Traceback (most recent call last):
...
ValueError: n must be at least 0
plot(xmin=None, xmax=None, *args, **kwds)

Return a plot of this polynomial.

INPUT:

  • xmin - float
  • xmax - float
  • *args, **kwds - passed to either plot or point

OUTPUT: returns a graphic object.

EXAMPLES:

sage: x = polygen(GF(389))
sage: plot(x^2 + 1, rgbcolor=(0,0,1))
sage: x = polygen(QQ)
sage: plot(x^2 + 1, rgbcolor=(1,0,0))
polynomial(var)

Let var be one of the variables of the parent of self. This returns self viewed as a univariate polynomial in var over the polynomial ring generated by all the other variables of the parent.

For univariate polynomials, if var is the generator of the parent ring, we return this polynomial, otherwise raise an error.

EXAMPLES:

sage: R.<x> = QQ[]
sage: (x+1).polynomial(x)
x + 1

TESTS:

sage: x.polynomial(1)
Traceback (most recent call last):
...
ValueError: given variable is not the generator of parent.
prec()

Return the precision of this polynomial. This is always infinity, since polynomials are of infinite precision by definition (there is no big-oh).

EXAMPLES:

sage: x = polygen(ZZ)
sage: (x^5 + x + 1).prec()
+Infinity
sage: x.prec()
+Infinity
radical()

Returns the radical of self; over a field, this is the product of the distinct irreducible factors of self. (This is also sometimes called the “square-free part” of self, but that term is ambiguous; it is sometimes used to mean the quotient of self by its maximal square factor.)

EXAMPLES:

sage: P.<x> = ZZ[]
sage: t = (x^2-x+1)^3 * (3*x-1)^2
sage: t.radical()
3*x^3 - 4*x^2 + 4*x - 1
sage: radical(12 * x^5)
6*x

If self has a factor of multiplicity divisible by the characteristic (see trac ticket #8736):

sage: P.<x> = GF(2)[]
sage: (x^3 + x^2).radical()
x^2 + x
real_roots()

Return the real roots of this polynomial, without multiplicities.

Calls self.roots(ring=RR), unless this is a polynomial with floating-point real coefficients, in which case it calls self.roots().

EXAMPLES:

sage: x = polygen(ZZ)
sage: (x^2 - x - 1).real_roots()
[-0.618033988749895, 1.61803398874989]

TESTS:

sage: x = polygen(RealField(100))
sage: (x^2 - x - 1).real_roots()[0].parent()
    Real Field with 100 bits of precision
sage: x = polygen(RDF)
sage: (x^2 - x - 1).real_roots()[0].parent()
Real Double Field

sage: x=polygen(ZZ,'x'); v=(x^2-x-1).real_roots()
sage: v[0].parent() is RR
True
resultant(other)

Returns the resultant of self and other.

INPUT:

  • other - a polynomial

OUTPUT: an element of the base ring of the polynomial ring

Note

Implemented using PARI’s polresultant function.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 + x + 1;  g = x^3 - x - 1
sage: r = f.resultant(g); r
-8
sage: r.parent() is QQ
True

We can also compute resultants over univariate and multivariate polynomial rings, provided that PARI’s variable ordering requirements are respected. Usually, your resultants will work if you always ask for them in the variable x:

sage: R.<a> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a; g = x^3 + a
sage: r = f.resultant(g); r
a^3 + a^2
sage: r.parent() is R
True
sage: R.<a, b> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a; g = x^3 + b
sage: r = f.resultant(g); r
a^3 + b^2
sage: r.parent() is R
True

Unfortunately Sage does not handle PARI’s variable ordering requirements gracefully, so the following has to be done through Singular:

sage: R.<x, y> = QQ[]
sage: S.<a> = R[]
sage: f = x^2 + a; g = y^3 + a
sage: h = f.resultant(g); h
y^3 - x^2
sage: h.parent() is R
True

Check that trac ticket #13672 is fixed:

sage: R.<t> = GF(2)[]
sage: S.<x> = R[]
sage: f = (t^2 + t)*x + t^2 + t
sage: g = (t + 1)*x + t^2
sage: f.resultant(g)
t^4 + t

Check that trac ticket #15061 is fixed:

sage: R.<T> = PowerSeriesRing(QQ)
sage: F = R([1,1],2)
sage: RP.<x> = PolynomialRing(R)
sage: P = x^2 - F
sage: P.resultant(P.derivative())
-4 - 4*T + O(T^2)

Check that trac ticket #16360 is fixed:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: y.resultant(y+x)
x

sage: K.<a> = FunctionField(QQ)
sage: R.<b> = K[]
sage: L.<b> = K.extension(b^2-a)
sage: R.<x> = L[]
sage: f=x^2-a
sage: g=x-b
sage: f.resultant(g)
0
reverse(degree=None)

Return polynomial but with the coefficients reversed.

If an optional degree argument is given the coefficient list will be truncated or zero padded as necessary and the reverse polynomial will have the specified degree.

EXAMPLES:

sage: R.<x> = ZZ[]; S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: f.reverse()
-3*x*y^3 + x*y^2 + 1
sage: f.reverse(degree=2)
-3*x*y^2 + x*y
sage: f.reverse(degree=5)
-3*x*y^5 + x*y^4 + y^2

TESTS:

sage: f.reverse(degree=1.5r)
Traceback (most recent call last):
...
ValueError: degree argument must be a non-negative integer, got 1.5
root_field(names, check_irreducible=True)

Return the field generated by the roots of the irreducible polynomial self. The output is either a number field, relative number field, a quotient of a polynomial ring over a field, or the fraction field of the base ring.

EXAMPLES:

sage: R.<x> = QQ['x']
sage: f = x^3 + x + 17
sage: f.root_field('a')
Number Field in a with defining polynomial x^3 + x + 17
sage: R.<x> = QQ['x']
sage: f = x - 3
sage: f.root_field('b')
Rational Field
sage: R.<x> = ZZ['x']
sage: f = x^3 + x + 17
sage: f.root_field('b')
Number Field in b with defining polynomial x^3 + x + 17
sage: y = QQ['x'].0
sage: L.<a> = NumberField(y^3-2)
sage: R.<x> = L['x']
sage: f = x^3 + x + 17
sage: f.root_field('c')
Number Field in c with defining polynomial x^3 + x + 17 over its base field
sage: R.<x> = PolynomialRing(GF(9,'a'))
sage: f = x^3 + x^2 + 8
sage: K.<alpha> = f.root_field(); K
Univariate Quotient Polynomial Ring in alpha over Finite Field in a of size 3^2 with modulus x^3 + x^2 + 2
sage: alpha^2 + 1
alpha^2 + 1
sage: alpha^3 + alpha^2
1
sage: R.<x> = QQ[]
sage: f = x^2
sage: K.<alpha> = f.root_field()
Traceback (most recent call last):
...
ValueError: polynomial must be irreducible

TESTS:

sage: (PolynomialRing(Integers(31),name='x').0+5).root_field('a')
Ring of integers modulo 31
roots(ring=None, multiplicities=True, algorithm=None)

Return the roots of this polynomial (by default, in the base ring of this polynomial).

INPUT:

  • ring - the ring to find roots in
  • multiplicities - bool (default: True) if True return list of pairs (r, n), where r is the root and n is the multiplicity. If False, just return the unique roots, with no information about multiplicities.
  • algorithm - the root-finding algorithm to use. We attempt to select a reasonable algorithm by default, but this lets the caller override our choice.

By default, this finds all the roots that lie in the base ring of the polynomial. However, the ring parameter can be used to specify a ring to look for roots in.

If the polynomial and the output ring are both exact (integers, rationals, finite fields, etc.), then the output should always be correct (or raise an exception, if that case is not yet handled).

If the output ring is approximate (floating-point real or complex numbers), then the answer will be estimated numerically, using floating-point arithmetic of at least the precision of the output ring. If the polynomial is ill-conditioned, meaning that a small change in the coefficients of the polynomial will lead to a relatively large change in the location of the roots, this may give poor results. Distinct roots may be returned as multiple roots, multiple roots may be returned as distinct roots, real roots may be lost entirely (because the numerical estimate thinks they are complex roots). Note that polynomials with multiple roots are always ill-conditioned; there’s a footnote at the end of the docstring about this.

If the output ring is a RealIntervalField or ComplexIntervalField of a given precision, then the answer will always be correct (or an exception will be raised, if a case is not implemented). Each root will be contained in one of the returned intervals, and the intervals will be disjoint. (The returned intervals may be of higher precision than the specified output ring.)

At the end of this docstring (after the examples) is a description of all the cases implemented in this function, and the algorithms used. That section also describes the possibilities for “algorithm=”, for the cases where multiple algorithms exist.

EXAMPLES:

sage: x = QQ['x'].0
sage: f = x^3 - 1
sage: f.roots()
[(1, 1)]
sage: f.roots(ring=CC)   # note -- low order bits slightly different on ppc.
[(1.00000000000000, 1), (-0.500000000000000 - 0.86602540378443...*I, 1), (-0.500000000000000 + 0.86602540378443...*I, 1)]
sage: f = (x^3 - 1)^2
sage: f.roots()
[(1, 2)]
sage: f = -19*x + 884736
sage: f.roots()
[(884736/19, 1)]
sage: (f^20).roots()
[(884736/19, 20)]
sage: K.<z> = CyclotomicField(3)
sage: f = K.defining_polynomial()
sage: f.roots(ring=GF(7))
[(4, 1), (2, 1)]
sage: g = f.change_ring(GF(7))
sage: g.roots()
[(4, 1), (2, 1)]
sage: g.roots(multiplicities=False)
[4, 2]

A new ring. In the example below, we add the special method _roots_univariate_polynomial to the base ring, and observe that this method is called instead to find roots of polynomials over this ring. This facility can be used to easily extend root finding to work over new rings you introduce:

sage: R.<x> = QQ[]
sage: (x^2 + 1).roots()
[]
sage: g = lambda f, *args, **kwds: f.change_ring(CDF).roots()
sage: QQ._roots_univariate_polynomial = g
sage: (x^2 + 1).roots()
[(... - 1.0*I, 1), (1.0*I, 1)]
sage: del QQ._roots_univariate_polynomial

An example over RR, which illustrates that only the roots in RR are returned:

sage: x = RR['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.25992104989487, 1)]
sage: f.factor()
(x - 1.25992104989487) * (x^2 + 1.25992104989487*x + 1.58740105196820)
sage: x = RealField(100)['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.2599210498948731647672106073, 1)]
sage: x = CC['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.25992104989487, 1), (-0.62996052494743... - 1.09112363597172*I, 1), (-0.62996052494743... + 1.09112363597172*I, 1)]
sage: f.roots(algorithm='pari')
[(1.25992104989487, 1), (-0.629960524947437 - 1.09112363597172*I, 1), (-0.629960524947437 + 1.09112363597172*I, 1)]

Another example showing that only roots in the base ring are returned:

sage: x = polygen(ZZ)
sage: f = (2*x-3) * (x-1) * (x+1)
sage: f.roots()
[(1, 1), (-1, 1)]
sage: f.roots(ring=QQ)
[(3/2, 1), (1, 1), (-1, 1)]

An example involving large numbers:

sage: x = RR['x'].0
sage: f = x^2 - 1e100
sage: f.roots()
[(-1.00000000000000e50, 1), (1.00000000000000e50, 1)]
sage: f = x^10 - 2*(5*x-1)^2
sage: f.roots(multiplicities=False)
[-1.6772670339941..., 0.19995479628..., 0.20004530611..., 1.5763035161844...]
sage: x = CC['x'].0
sage: i = CC.0
sage: f = (x - 1)*(x - i)
sage: f.roots(multiplicities=False) #random - this example is numerically rather unstable
[2.22044604925031e-16 + 1.00000000000000*I, 1.00000000000000 + 8.32667268468867e-17*I]
sage: g=(x-1.33+1.33*i)*(x-2.66-2.66*i)
sage: g.roots(multiplicities=False)
[1.33000000000000 - 1.33000000000000*I, 2.66000000000000 + 2.66000000000000*I]

A purely symbolic roots example:

sage: X = var('X')
sage: f = expand((X-1)*(X-I)^3*(X^2 - sqrt(2))); f
X^6 - (3*I + 1)*X^5 - sqrt(2)*X^4 + (3*I - 3)*X^4 + (3*I + 1)*sqrt(2)*X^3 + (I + 3)*X^3 - (3*I - 3)*sqrt(2)*X^2 - I*X^2 - (I + 3)*sqrt(2)*X + I*sqrt(2)
sage: print f.roots()
[(I, 3), (-2^(1/4), 1), (2^(1/4), 1), (1, 1)]

A couple of examples where the base ring doesn’t have a factorization algorithm (yet). Note that this is currently done via naive enumeration, so could be very slow:

sage: R = Integers(6)
sage: S.<x> = R['x']
sage: p = x^2-1
sage: p.roots()
Traceback (most recent call last):
...
NotImplementedError: root finding with multiplicities for this polynomial not implemented (try the multiplicities=False option)
sage: p.roots(multiplicities=False)
[1, 5]
sage: R = Integers(9)
sage: A = PolynomialRing(R, 'y')
sage: y = A.gen()
sage: f = 10*y^2 - y^3 - 9
sage: f.roots(multiplicities=False)
[0, 1, 3, 6]

An example over the complex double field (where root finding is fast, thanks to NumPy):

sage: R.<x> = CDF[]
sage: f = R.cyclotomic_polynomial(5); f
x^4 + x^3 + x^2 + x + 1.0
sage: f.roots(multiplicities=False)   # slightly random
[0.309016994375 + 0.951056516295*I, 0.309016994375 - 0.951056516295*I, -0.809016994375 + 0.587785252292*I, -0.809016994375 - 0.587785252292*I]
sage: [z^5 for z in f.roots(multiplicities=False)]     # slightly random
[1.0 - 2.44929359829e-16*I, 1.0 + 2.44929359829e-16*I, 1.0 - 4.89858719659e-16*I, 1.0 + 4.89858719659e-16*I]
sage: f = CDF['x']([1,2,3,4]); f
4.0*x^3 + 3.0*x^2 + 2.0*x + 1.0
sage: r = f.roots(multiplicities=False)
sage: [f(a) for a in r]    # slightly random
[2.55351295664e-15, -4.4408920985e-16 - 2.08166817117e-16*I, -4.4408920985e-16 + 2.08166817117e-16*I]

Another example over RDF:

sage: x = RDF['x'].0
sage: ((x^3 -1)).roots()
[(1.0, 1)]
sage: ((x^3 -1)).roots(multiplicities=False)
[1.0]

More examples involving the complex double field:

sage: x = CDF['x'].0
sage: i = CDF.0
sage: f = x^3 + 2*i; f
x^3 + 2.0*I
sage: f.roots()
[(-1.09112363597 - 0.629960524947*I, 1),
 (... + 1.25992104989*I, 1),
 (1.09112363597 - 0.629960524947*I, 1)]
sage: f.roots(multiplicities=False)
[-1.09112363597 - 0.629960524947*I, ... + 1.25992104989*I, 1.09112363597 - 0.629960524947*I]
sage: [f(z) for z in f.roots(multiplicities=False)] # random, too close to 0
[-2.56337823492e-15 - 6.66133814775e-15*I,
 3.96533069372e-16 + 1.99840144433e-15*I,
 4.19092485179e-17 - 8.881784197e-16*I]
sage: f = i*x^3 + 2; f
I*x^3 + 2.0
sage: f.roots()
[(-1.09112363597 + 0.629960524947*I, 1),
 (... - 1.25992104989*I, 1),
 (1.09112363597 + 0.629960524947*I, 1)]
sage: f(f.roots()[0][0]) # random, too close to 0
-2.56337823492e-15 - 6.66133814775e-15*I

Examples using real root isolation:

sage: x = polygen(ZZ)
sage: f = x^2 - x - 1
sage: f.roots()
[]
sage: f.roots(ring=RIF)
[(-0.6180339887498948482045868343657?, 1), (1.6180339887498948482045868343657?, 1)]
sage: f.roots(ring=RIF, multiplicities=False)
[-0.6180339887498948482045868343657?, 1.6180339887498948482045868343657?]
sage: f.roots(ring=RealIntervalField(150))
[(-0.6180339887498948482045868343656381177203091798057628621354486227?, 1), (1.618033988749894848204586834365638117720309179805762862135448623?, 1)]
sage: f.roots(ring=AA)
[(-0.618033988749895?, 1), (1.618033988749895?, 1)]
sage: f = f^2 * (x - 1)
sage: f.roots(ring=RIF)
[(-0.6180339887498948482045868343657?, 2), (1.0000000000000000000000000000000?, 1), (1.6180339887498948482045868343657?, 2)]
sage: f.roots(ring=RIF, multiplicities=False)
[-0.6180339887498948482045868343657?, 1.0000000000000000000000000000000?, 1.6180339887498948482045868343657?]

Examples using complex root isolation:

sage: x = polygen(ZZ)
sage: p = x^5 - x - 1
sage: p.roots()
[]
sage: p.roots(ring=CIF)
[(1.167303978261419?, 1), (-0.764884433600585? - 0.352471546031727?*I, 1), (-0.764884433600585? + 0.352471546031727?*I, 1), (0.181232444469876? - 1.083954101317711?*I, 1), (0.181232444469876? + 1.083954101317711?*I, 1)]
sage: p.roots(ring=ComplexIntervalField(200))
[(1.167303978261418684256045899854842180720560371525489039140082?, 1), (-0.76488443360058472602982318770854173032899665194736756700778? - 0.35247154603172624931794709140258105439420648082424733283770?*I, 1), (-0.76488443360058472602982318770854173032899665194736756700778? + 0.35247154603172624931794709140258105439420648082424733283770?*I, 1), (0.18123244446987538390180023778112063996871646618462304743774? - 1.08395410131771066843034449298076657427364024315511565430114?*I, 1), (0.18123244446987538390180023778112063996871646618462304743774? + 1.08395410131771066843034449298076657427364024315511565430114?*I, 1)]
sage: rts = p.roots(ring=QQbar); rts
[(1.167303978261419?, 1), (-0.7648844336005847? - 0.3524715460317263?*I, 1), (-0.7648844336005847? + 0.3524715460317263?*I, 1), (0.1812324444698754? - 1.083954101317711?*I, 1), (0.1812324444698754? + 1.083954101317711?*I, 1)]
sage: p.roots(ring=AA)
[(1.167303978261419?, 1)]
sage: p = (x - rts[4][0])^2 * (3*x^2 + x + 1)
sage: p.roots(ring=QQbar)
[(-0.1666666666666667? - 0.552770798392567?*I, 1), (-0.1666666666666667? + 0.552770798392567?*I, 1), (0.1812324444698754? + 1.083954101317711?*I, 2)]
sage: p.roots(ring=CIF)
[(-0.1666666666666667? - 0.552770798392567?*I, 1), (-0.1666666666666667? + 0.552770798392567?*I, 1), (0.1812324444698754? + 1.083954101317711?*I, 2)]

Note that coefficients in a number field with defining polynomial \(x^2 + 1\) are considered to be Gaussian rationals (with the generator mapping to +I), if you ask for complex roots.

sage: K.<im> = NumberField(x^2 + 1)
sage: y = polygen(K)
sage: p = y^4 - 2 - im
sage: p.roots(ring=CC)
[(-1.2146389322441... - 0.14142505258239...*I, 1), (-0.14142505258239... + 1.2146389322441...*I, 1), (0.14142505258239... - 1.2146389322441...*I, 1), (1.2146389322441... + 0.14142505258239...*I, 1)]
sage: p = p^2 * (y^2 - 2)
sage: p.roots(ring=CIF)
[(-1.414213562373095?, 1), (1.414213562373095?, 1), (-1.214638932244183? - 0.141425052582394?*I, 2), (-0.141425052582394? + 1.214638932244183?*I, 2), (0.141425052582394? - 1.214638932244183?*I, 2), (1.214638932244183? + 0.141425052582394?*I, 2)]

Note that one should not use NumPy when wanting high precision output as it does not support any of the high precision types:

sage: R.<x> = RealField(200)[]
sage: f = x^2 - R(pi)
sage: f.roots()
[(-1.7724538509055160272981674833411451827975494561223871282138, 1), (1.7724538509055160272981674833411451827975494561223871282138, 1)]
sage: f.roots(algorithm='numpy')
doctest... UserWarning: NumPy does not support arbitrary precision arithmetic.  The roots found will likely have less precision than you expect.
[(-1.77245385090551..., 1), (1.77245385090551..., 1)]

We can also find roots over number fields:

sage: K.<z> = CyclotomicField(15)
sage: R.<x> = PolynomialRing(K)
sage: (x^2 + x + 1).roots()
[(z^5, 1), (-z^5 - 1, 1)]

There are many combinations of floating-point input and output types that work. (Note that some of them are quite pointless like using algorithm='numpy' with high-precision types.)

sage: rflds = (RR, RDF, RealField(100))
sage: cflds = (CC, CDF, ComplexField(100))
sage: def cross(a, b):
...       return list(cartesian_product_iterator([a, b]))
sage: flds = cross(rflds, rflds) + cross(rflds, cflds) + cross(cflds, cflds)
sage: for (fld_in, fld_out) in flds:
...       x = polygen(fld_in)
...       f = x^3 - fld_in(2)
...       x2 = polygen(fld_out)
...       f2 = x2^3 - fld_out(2)
...       for algo in (None, 'pari', 'numpy'):
...           rts = f.roots(ring=fld_out, multiplicities=False)
...           if fld_in == fld_out and algo is None:
...               print fld_in, rts
...           for rt in rts:
...               assert(abs(f2(rt)) <= 1e-10)
...               assert(rt.parent() == fld_out)
Real Field with 53 bits of precision [1.25992104989487]
Real Double Field [1.25992104989]
Real Field with 100 bits of precision [1.2599210498948731647672106073]
Complex Field with 53 bits of precision [1.25992104989487, -0.62996052494743... - 1.09112363597172*I, -0.62996052494743... + 1.09112363597172*I]
Complex Double Field [1.25992104989, -0.62996052494... - 1.09112363597*I, -0.62996052494... + 1.09112363597*I]
Complex Field with 100 bits of precision [1.2599210498948731647672106073, -0.62996052494743658238360530364 - 1.0911236359717214035600726142*I, -0.62996052494743658238360530364 + 1.0911236359717214035600726142*I]

Note that we can find the roots of a polynomial with algebraic coefficients:

sage: rt2 = sqrt(AA(2))
sage: rt3 = sqrt(AA(3))
sage: x = polygen(AA)
sage: f = (x - rt2) * (x - rt3); f
    x^2 - 3.146264369941973?*x + 2.449489742783178?
sage: rts = f.roots(); rts
[(1.414213562373095?, 1), (1.732050807568878?, 1)]
sage: rts[0][0] == rt2
True
sage: f.roots(ring=RealIntervalField(150))
[(1.414213562373095048801688724209698078569671875376948073176679738?, 1), (1.732050807568877293527446341505872366942805253810380628055806980?, 1)]

We can handle polynomials with huge coefficients.

This number doesn’t even fit in an IEEE double-precision float, but RR and CC allow a much larger range of floating-point numbers:

sage: bigc = 2^1500
sage: CDF(bigc)
+infinity
sage: CC(bigc)
3.50746621104340e451

Polynomials using such large coefficients can’t be handled by numpy, but pari can deal with them:

sage: x = polygen(QQ)
sage: p = x + bigc
sage: p.roots(ring=RR, algorithm='numpy')
Traceback (most recent call last):
...
LinAlgError: Array must not contain infs or NaNs
sage: p.roots(ring=RR, algorithm='pari')
[(-3.50746621104340e451, 1)]
sage: p.roots(ring=AA)
[(-3.5074662110434039?e451, 1)]
sage: p.roots(ring=QQbar)
[(-3.5074662110434039?e451, 1)]
sage: p = bigc*x + 1
sage: p.roots(ring=RR)
[(0.000000000000000, 1)]
sage: p.roots(ring=AA)
[(-2.8510609648967059?e-452, 1)]
sage: p.roots(ring=QQbar)
[(-2.8510609648967059?e-452, 1)]
sage: p = x^2 - bigc
sage: p.roots(ring=RR)
[(-5.92238652153286e225, 1), (5.92238652153286e225, 1)]
sage: p.roots(ring=QQbar)
[(-5.9223865215328558?e225, 1), (5.9223865215328558?e225, 1)]

Algorithms used:

For brevity, we will use RR to mean any RealField of any precision; similarly for RIF, CC, and CIF. Since Sage has no specific implementation of Gaussian rationals (or of number fields with embedding, at all), when we refer to Gaussian rationals below we will accept any number field with defining polynomial \(x^2+1\), mapping the field generator to +I.

We call the base ring of the polynomial K, and the ring given by the ring= argument L. (If ring= is not specified, then L is the same as K.)

If K and L are floating-point (RDF, CDF, RR, or CC), then a floating-point root-finder is used. If L is RDF or CDF then we default to using NumPy’s roots(); otherwise, we use PARI’s polroots(). This choice can be overridden with algorithm=’pari’ or algorithm=’numpy’. If the algorithm is unspecified and NumPy’s roots() algorithm fails, then we fall back to pari (numpy will fail if some coefficient is infinite, for instance).

If L is AA or RIF, and K is ZZ, QQ, or AA, then the root isolation algorithm sage.rings.polynomial.real_roots.real_roots() is used. (You can call real_roots() directly to get more control than this method gives.)

If L is QQbar or CIF, and K is ZZ, QQ, AA, QQbar, or the Gaussian rationals, then the root isolation algorithm sage.rings.polynomial.complex_roots.complex_roots() is used. (You can call complex_roots() directly to get more control than this method gives.)

If L is AA and K is QQbar or the Gaussian rationals, then complex_roots() is used (as above) to find roots in QQbar, then these roots are filtered to select only the real roots.

If L is floating-point and K is not, then we attempt to change the polynomial ring to L (using .change_ring()) (or, if L is complex and K is not, to the corresponding real field). Then we use either PARI or numpy as specified above.

For all other cases where K is different than L, we just use .change_ring(L) and proceed as below.

The next method, which is used if K is an integral domain, is to attempt to factor the polynomial. If this succeeds, then for every degree-one factor a*x+b, we add -b/a as a root (as long as this quotient is actually in the desired ring).

If factoring over K is not implemented (or K is not an integral domain), and K is finite, then we find the roots by enumerating all elements of K and checking whether the polynomial evaluates to zero at that value.

Note

We mentioned above that polynomials with multiple roots are always ill-conditioned; if your input is given to n bits of precision, you should not expect more than n/k good bits for a k-fold root. (You can get solutions that make the polynomial evaluate to a number very close to zero; basically the problem is that with a multiple root, there are many such numbers, and it’s difficult to choose between them.)

To see why this is true, consider the naive floating-point error analysis model where you just pretend that all floating-point numbers are somewhat imprecise - a little ‘fuzzy’, if you will. Then the graph of a floating-point polynomial will be a fuzzy line. Consider the graph of \((x-1)^3\); this will be a fuzzy line with a horizontal tangent at \(x=1\), \(y=0\). If the fuzziness extends up and down by about j, then it will extend left and right by about cube_root(j).

TESTS:

sage: K.<zeta> = CyclotomicField(2)
sage: R.<x> = K[]
sage: factor(x^3-1)
(x - 1) * (x^2 + x + 1)

This shows that the issue from trac ticket #6237 is fixed:

sage: R.<u> = QQ[]
sage: g = -27*u^14 - 32*u^9
sage: g.roots(CDF, multiplicities=False)
[-1.03456371594, 0.0, -0.31969776999 - 0.983928563571*I, -0.31969776999 + 0.983928563571*I, 0.836979627962 - 0.608101294789*I, 0.836979627962 + 0.608101294789*I]
sage: g.roots(CDF)
[(-1.03456371594, 1), (0.0, 9), (-0.31969776999 - 0.983928563571*I, 1), (-0.31969776999 + 0.983928563571*I, 1), (0.836979627962 - 0.608101294789*I, 1), (0.836979627962 + 0.608101294789*I, 1)]

This shows that the issue at trac ticket #2418 is fixed:

sage: x = polygen(QQ)
sage: p = (x^50/2^100 + x^10 + x + 1).change_ring(ComplexField(106))
sage: rts = (p/2^100).roots(multiplicities=False)
sage: eps = 2^(-50)   # we test the roots numerically
sage: [abs(p(rt)) < eps for rt in rts] == [True]*50
True

This shows that the issue at trac ticket #10901 is fixed:

sage: a = var('a'); R.<x> = SR[]
sage: f = x - a
sage: f.roots(RR)
Traceback (most recent call last):
...
TypeError: Cannot evaluate symbolic expression to a numeric value.
sage: f.roots(CC)
Traceback (most recent call last):
...
TypeError: Cannot evaluate symbolic expression to a numeric value.

We can find roots of polynomials defined over \(\ZZ\) or \(\QQ\) over the \(p\)-adics, see trac ticket #15422:

sage: R.<x> = ZZ[]
sage: pol = (x - 1)^2
sage: pol.roots(Qp(3,5))
[(1 + O(3^5), 2)]

This doesn’t work if we first change coefficients to \(\QQ_p\):

sage: pol.change_ring(Qp(3,5)).roots()
Traceback (most recent call last):
...
PrecisionError: p-adic factorization not well-defined since the discriminant is zero up to the requestion p-adic precision

sage: (pol - 3^6).roots(Qp(3,5))
[(1 + 2*3^3 + 2*3^4 + O(3^5), 1), (1 + 3^3 + O(3^5), 1)]
sage: r = pol.roots(Zp(3,5), multiplicities=False); r
[1 + O(3^5)]
sage: parent(r[0])
3-adic Ring with capped relative precision 5
shift(n)

Returns this polynomial multiplied by the power \(x^n\). If \(n\) is negative, terms below \(x^n\) will be discarded. Does not change this polynomial (since polynomials are immutable).

EXAMPLES:

sage: R.<x> = QQ[]
sage: p = x^2 + 2*x + 4
sage: p.shift(0)
 x^2 + 2*x + 4
sage: p.shift(-1)
 x + 2
sage: p.shift(-5)
 0
sage: p.shift(2)
 x^4 + 2*x^3 + 4*x^2

One can also use the infix shift operator:

sage: f = x^3 + x
sage: f >> 2
x
sage: f << 2
x^5 + x^3

TESTS:

sage: p = R(0)
sage: p.shift(3).is_zero()
True
sage: p.shift(-3).is_zero()
True

AUTHORS:

  • David Harvey (2006-08-06)
  • Robert Bradshaw (2007-04-18): Added support for infix operator.
splitting_field(names, map=False, **kwds)

Compute the absolute splitting field of a given polynomial.

INPUT:

  • names – a variable name for the splitting field.
  • map – (default: False) also return an embedding of self into the resulting field.
  • kwds – additional keywords depending on the type. Currently, only number fields are implemented. See sage.rings.number_field.splitting_field.splitting_field() for the documentation of these keywords.

OUTPUT:

If map is False, the splitting field as an absolute field. If map is True, a tuple (K, phi) where phi is an embedding of the base field of self in K.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
sage: K.<a> = (x^3 + 2).splitting_field(); K
Number Field in a with defining polynomial x^6 + 3*x^5 + 6*x^4 + 11*x^3 + 12*x^2 - 3*x + 1
sage: K.<a> = (x^3 - 3*x + 1).splitting_field(); K
Number Field in a with defining polynomial x^3 - 3*x + 1

Relative situation:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 + 2)
sage: S.<t> = PolynomialRing(K)
sage: L.<b> = (t^2 - a).splitting_field()
sage: L
Number Field in b with defining polynomial t^6 + 2

With map=True, we also get the embedding of the base field into the splitting field:

sage: L.<b>, phi = (t^2 - a).splitting_field(map=True)
sage: phi
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Number Field in b with defining polynomial t^6 + 2
  Defn: a |--> b^2

An example over a finite field:

sage: P.<x> = PolynomialRing(GF(7))
sage: t = x^2 + 1
sage: t.splitting_field('b')
Finite Field in b of size 7^2

sage: P.<x> = PolynomialRing(GF(7^3, 'a'))
sage: t = x^2 + 1
sage: t.splitting_field('b', map=True)
(Finite Field in b of size 7^6,
 Ring morphism:
   From: Finite Field in a of size 7^3
   To:   Finite Field in b of size 7^6
   Defn: a |--> 4*b^5 + 4*b^4 + 4*b^3 + 2*b^2 + 4*b + 5)

If the extension is trivial and the generators have the same name, the map will be the identity:

sage: t = 24*x^13 + 2*x^12 + 14
sage: t.splitting_field('a', map=True)
(Finite Field in a of size 7^3,
 Identity endomorphism of Finite Field in a of size 7^3)

sage: t = x^56 - 14*x^3
sage: t.splitting_field('b', map=True)
(Finite Field in b of size 7^3,
 Ring morphism:
 From: Finite Field in a of size 7^3
   To:   Finite Field in b of size 7^3
   Defn: a |--> b)

See also

sage.rings.number_field.splitting_field.splitting_field() for more examples over number fields

TESTS:

sage: K.<a,b> = x.splitting_field()
Traceback (most recent call last):
...
IndexError: the number of names must equal the number of generators
sage: polygen(RR).splitting_field('x')
Traceback (most recent call last):
...
NotImplementedError: splitting_field() is only implemented over number fields and finite fields

sage: P.<x> = PolynomialRing(GF(11^5, 'a'))
sage: t = x^2 + 1
sage: t.splitting_field('b')
Finite Field in b of size 11^10
sage: t = 24*x^13 + 2*x^12 + 14
sage: t.splitting_field('b')
Finite Field in b of size 11^30
sage: t = x^56 - 14*x^3
sage: t.splitting_field('b')
Finite Field in b of size 11^130

sage: P.<x> = PolynomialRing(GF(19^6, 'a'))
sage: t = -x^6 + x^2 + 1
sage: t.splitting_field('b')
Finite Field in b of size 19^6
sage: t = 24*x^13 + 2*x^12 + 14
sage: t.splitting_field('b')
Finite Field in b of size 19^18
sage: t = x^56 - 14*x^3
sage: t.splitting_field('b')
Finite Field in b of size 19^156

sage: P.<x> = PolynomialRing(GF(83^6, 'a'))
sage: t = 2*x^14 - 5 + 6*x
sage: t.splitting_field('b')
Finite Field in b of size 83^84
sage: t = 24*x^13 + 2*x^12 + 14
sage: t.splitting_field('b')
Finite Field in b of size 83^78
sage: t = x^56 - 14*x^3
sage: t.splitting_field('b')
Finite Field in b of size 83^12

sage: P.<x> = PolynomialRing(GF(401^13, 'a'))
sage: t = 2*x^14 - 5 + 6*x
sage: t.splitting_field('b')
Finite Field in b of size 401^104
sage: t = 24*x^13 + 2*x^12 + 14
sage: t.splitting_field('b')
Finite Field in b of size 401^156
sage: t = x^56 - 14*x^3
sage: t.splitting_field('b')
Finite Field in b of size 401^52
square()

Returns the square of this polynomial.

TODO:

  • This is just a placeholder; for now it just uses ordinary multiplication. But generally speaking, squaring is faster than ordinary multiplication, and it’s frequently used, so subclasses may choose to provide a specialised squaring routine.
  • Perhaps this even belongs at a lower level? ring_element or something?

AUTHORS:

  • David Harvey (2006-09-09)

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 + 1
sage: f.square()
x^6 + 2*x^3 + 1
sage: f*f
x^6 + 2*x^3 + 1
squarefree_decomposition()

Return the square-free decomposition of this polynomial. This is a partial factorization into square-free, coprime polynomials.

EXAMPLES:

sage: x = polygen(QQ)
sage: p = 37 * (x-1)^3 * (x-2)^3 * (x-1/3)^7 * (x-3/7)
sage: p.squarefree_decomposition()
(37*x - 111/7) * (x^2 - 3*x + 2)^3 * (x - 1/3)^7
sage: p = 37 * (x-2/3)^2
sage: p.squarefree_decomposition()
(37) * (x - 2/3)^2
sage: x = polygen(GF(3))
sage: x.squarefree_decomposition()
x
sage: f = QQbar['x'](1)
sage: f.squarefree_decomposition()
1
subs(*x, **kwds)

Identical to self(*x).

See the docstring for self.__call__.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 + x - 3
sage: f.subs(x=5)
127
sage: f.subs(5)
127
sage: f.subs({x:2})
7
sage: f.subs({})
x^3 + x - 3
sage: f.subs({'x':2})
Traceback (most recent call last):
...
TypeError: keys do not match self's parent
substitute(*x, **kwds)

Identical to self(*x).

See the docstring for self.__call__.

EXAMPLES:

sage: R.<x> = QQ[]
sage: f = x^3 + x - 3
sage: f.subs(x=5)
127
sage: f.subs(5)
127
sage: f.subs({x:2})
7
sage: f.subs({})
x^3 + x - 3
sage: f.subs({'x':2})
Traceback (most recent call last):
...
TypeError: keys do not match self's parent
sylvester_matrix(right, variable=None)

Returns the Sylvester matrix of self and right.

Note that the Sylvester matrix is not defined if one of the polynomials is zero.

INPUT:

  • right: a polynomial in the same ring as self.
  • variable: optional, included for compatibility with the multivariate case only. The variable of the polynomials.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
sage: f = (6*x + 47)*(7*x^2 - 2*x + 38)
sage: g = (6*x + 47)*(3*x^3 + 2*x + 1)
sage: M = f.sylvester_matrix(g)
sage: M
[  42  317  134 1786    0    0    0]
[   0   42  317  134 1786    0    0]
[   0    0   42  317  134 1786    0]
[   0    0    0   42  317  134 1786]
[  18  141   12  100   47    0    0]
[   0   18  141   12  100   47    0]
[   0    0   18  141   12  100   47]

If the polynomials share a non-constant common factor then the determinant of the Sylvester matrix will be zero:

sage: M.determinant()
0

If self and right are polynomials of positive degree, the determinant of the Sylvester matrix is the resultant of the polynomials.:

sage: h1 = R.random_element()
sage: h2 = R.random_element()
sage: M1 = h1.sylvester_matrix(h2)
sage: M1.determinant() == h1.resultant(h2)
True

The rank of the Sylvester matrix is related to the degree of the gcd of self and right:

sage: f.gcd(g).degree() == f.degree() + g.degree() - M.rank()
True
sage: h1.gcd(h2).degree() == h1.degree() + h2.degree() - M1.rank()
True

TESTS:

The variable is optional, but must be the same in both rings:

sage: K.<x> = QQ['x']
sage: f = x+1
sage: g = QQ['y']([1, 0, 1])
sage: f.sylvester_matrix(f, x)
[1 1]
[1 1]
sage: f.sylvester_matrix(g, x)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents: 'Univariate Polynomial Ring in x over Rational Field' and 'Univariate Polynomial Ring in y over Rational Field'

Polynomials must be defined over compatible base rings:

sage: f = QQ['x']([1, 0, 1])
sage: g = ZZ['x']([1, 0, 1])
sage: h = GF(25, 'a')['x']([1, 0, 1])
sage: f.sylvester_matrix(g)
[1 0 1 0]
[0 1 0 1]
[1 0 1 0]
[0 1 0 1]
sage: g.sylvester_matrix(h)
[1 0 1 0]
[0 1 0 1]
[1 0 1 0]
[0 1 0 1]
sage: f.sylvester_matrix(h)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents: 'Univariate Polynomial Ring in x over Rational Field' and 'Univariate Polynomial Ring in x over Finite Field in a of size 5^2'

We can compute the sylvester matrix of a univariate and multivariate polynomial:

sage: K.<x,y> = QQ['x,y']
sage: g = K.random_element()
sage: f.sylvester_matrix(g) == K(f).sylvester_matrix(g,x)
True

Corner cases:

sage: K.<x>=QQ[]
sage: f = x^2+1
sage: g = K(0)
sage: f.sylvester_matrix(g)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: g.sylvester_matrix(f)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: g.sylvester_matrix(g)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: K(3).sylvester_matrix(x^2)
[3 0]
[0 3]
sage: K(3).sylvester_matrix(K(4))
[]
truncate(n)

Returns the polynomial of degree ` < n` which is equivalent to self modulo \(x^n\).

EXAMPLES:

sage: R.<x> = ZZ[]; S.<y> = PolynomialRing(R, sparse=True)
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: f.truncate(2)
x*y - 3*x
sage: f.truncate(1)
-3*x
sage: f.truncate(0)
0
valuation(p=None)

If \(f = a_r x^r + a_{r+1}x^{r+1} + \cdots\), with \(a_r\) nonzero, then the valuation of \(f\) is \(r\). The valuation of the zero polynomial is \(\infty\).

If a prime (or non-prime) \(p\) is given, then the valuation is the largest power of \(p\) which divides self.

The valuation at \(\infty\) is -self.degree().

EXAMPLES:

sage: P,x=PolynomialRing(ZZ,'x').objgen()
sage: (x^2+x).valuation()
1
sage: (x^2+x).valuation(x+1)
1
sage: (x^2+1).valuation()
0
sage: (x^3+1).valuation(infinity)
-3
sage: P(0).valuation()
+Infinity
variable_name()

Return name of variable used in this polynomial as a string.

OUTPUT: string

EXAMPLES:

sage: R.<t> = QQ[]
sage: f = t^3 + 3/2*t + 5
sage: f.variable_name()
't'
variables()

Returns the tuple of variables occurring in this polynomial.

EXAMPLES:

sage: R.<x> = QQ[]
sage: x.variables()
(x,)

A constant polynomial has no variables.

sage: R(2).variables()
()
class sage.rings.polynomial.polynomial_element.PolynomialBaseringInjection

Bases: sage.categories.morphism.Morphism

This class is used for conversion from a ring to a polynomial over that ring.

It calls the _new_constant_poly method on the generator, which should be optimized for a particular polynomial type.

Technically, it should be a method of the polynomial ring, but few polynomial rings are cython classes, and so, as a method of a cython polynomial class, it is faster.

EXAMPLES:

We demonstrate that most polynomial ring classes use polynomial base injection maps for coercion. They are supposed to be the fastest maps for that purpose. See trac ticket #9944.

sage: R.<x> = Qp(3)[]
sage: R.coerce_map_from(R.base_ring())
Polynomial base injection morphism:
  From: 3-adic Field with capped relative precision 20
  To:   Univariate Polynomial Ring in x over 3-adic Field with capped relative precision 20
sage: R.<x,y> = Qp(3)[]
sage: R.coerce_map_from(R.base_ring())
Polynomial base injection morphism:
  From: 3-adic Field with capped relative precision 20
  To:   Multivariate Polynomial Ring in x, y over 3-adic Field with capped relative precision 20
sage: R.<x,y> = QQ[]
sage: R.coerce_map_from(R.base_ring())
Polynomial base injection morphism:
  From: Rational Field
  To:   Multivariate Polynomial Ring in x, y over Rational Field
sage: R.<x> = QQ[]
sage: R.coerce_map_from(R.base_ring())
Polynomial base injection morphism:
  From: Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field

By trac ticket #9944, there are now only very few exceptions:

sage: PolynomialRing(QQ,names=[]).coerce_map_from(QQ)
Generic morphism:
  From: Rational Field
  To:   Multivariate Polynomial Ring in no variables over Rational Field
section()

TESTS:

sage: from sage.rings.polynomial.polynomial_element import PolynomialBaseringInjection
sage: m = PolynomialBaseringInjection(RDF, RDF['x'])
sage: m.section()
Generic map:
  From: Univariate Polynomial Ring in x over Real Double Field
  To:   Real Double Field
sage: type(m.section())
<type 'sage.rings.polynomial.polynomial_element.ConstantPolynomialSection'>
class sage.rings.polynomial.polynomial_element.Polynomial_generic_dense

Bases: sage.rings.polynomial.polynomial_element.Polynomial

A generic dense polynomial.

EXAMPLES:

sage: R.<x> = PolynomialRing(PolynomialRing(QQ,'y'))
sage: f = x^3 - x + 17
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: loads(f.dumps()) == f
True
constant_coefficient()

Return the constant coefficient of this polynomial.

OUTPUT:
element of base ring
EXAMPLES:
sage: R.<t> = QQ[] sage: S.<x> = R[] sage: f = x*t + x + t sage: f.constant_coefficient() t
degree(gen=None)

EXAMPLES:

sage: R.<x> = RDF[]
sage: f = (1+2*x^7)^5
sage: f.degree()
35

TESTS:

Check that trac ticket #12552 is fixed:

sage: type(f.degree())
<type 'sage.rings.integer.Integer'>
list(copy=True)

Return a new copy of the list of the underlying elements of self.

EXAMPLES:

sage: R.<x> = GF(17)[]
sage: f = (1+2*x)^3 + 3*x; f
8*x^3 + 12*x^2 + 9*x + 1
sage: f.list()
[1, 9, 12, 8]
quo_rem(other)

Returns the quotient and remainder of the Euclidean division of self and other.

Raises ZerodivisionError if other is zero. Raises ArithmeticError if other has a nonunit leading coefficient.

EXAMPLES:

sage: P.<x> = QQ[]
sage: R.<y> = P[]
sage: f = R.random_element(10)
sage: g = y^5+R.random_element(4)
sage: q,r = f.quo_rem(g)
sage: f == q*g + r
True
sage: g = x*y^5
sage: f.quo_rem(g)
Traceback (most recent call last):
...
ArithmeticError: Nonunit leading coefficient
sage: g = 0
sage: f.quo_rem(g)
Traceback (most recent call last):
...
ZeroDivisionError: Division by zero polynomial
shift(n)

Returns this polynomial multiplied by the power \(x^n\). If \(n\) is negative, terms below \(x^n\) will be discarded. Does not change this polynomial.

EXAMPLES:

sage: R.<x> = PolynomialRing(PolynomialRing(QQ,'y'), 'x')
sage: p = x^2 + 2*x + 4
sage: type(p)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: p.shift(0)
 x^2 + 2*x + 4
sage: p.shift(-1)
 x + 2
sage: p.shift(2)
 x^4 + 2*x^3 + 4*x^2

TESTS:

sage: p = R(0)
sage: p.shift(3).is_zero()
True
sage: p.shift(-3).is_zero()
True

AUTHORS:

  • David Harvey (2006-08-06)
truncate(n)

Returns the polynomial of degree ` < n` which is equivalent to self modulo \(x^n\).

EXAMPLES:

sage: S.<q> = QQ['t']['q']
sage: f = (1+q^10+q^11+q^12).truncate(11); f
q^10 + 1
sage: f = (1+q^10+q^100).truncate(50); f
q^10 + 1
sage: f.degree()
10
sage: f = (1+q^10+q^100).truncate(500); f
q^100 + q^10 + 1

TESTS:

Make sure we’re not actually testing a specialized implementation.

sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage.rings.polynomial.polynomial_element.is_Polynomial(f)

Return True if f is of type univariate polynomial.

INPUT:

  • f - an object

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_element import is_Polynomial
sage: R.<x> = ZZ[]
sage: is_Polynomial(x^3 + x + 1)
True
sage: S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
True

However this function does not return True for genuine multivariate polynomial type objects or symbolic polynomials, since those are not of the same data type as univariate polynomials:

sage: R.<x,y> = QQ[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
False
sage: var('x,y')
(x, y)
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
False
sage.rings.polynomial.polynomial_element.make_generic_polynomial(parent, coeffs)
sage.rings.polynomial.polynomial_element.universal_discriminant(n)

Return the discriminant of the ‘universal’ univariate polynomial \(a_n x^n + \cdots + a_1 x + a_0\) in \(\ZZ[a_0, \ldots, a_n][x]\).

INPUT:

  • n - degree of the polynomial

OUTPUT:

The discriminant as a polynomial in \(n + 1\) variables over \(\ZZ\). The result will be cached, so subsequent computations of discriminants of the same degree will be faster.

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_element import universal_discriminant
sage: universal_discriminant(1)
1
sage: universal_discriminant(2)
a1^2 - 4*a0*a2
sage: universal_discriminant(3)
a1^2*a2^2 - 4*a0*a2^3 - 4*a1^3*a3 + 18*a0*a1*a2*a3 - 27*a0^2*a3^2
sage: universal_discriminant(4).degrees()
(3, 4, 4, 4, 3)

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