Univariate Polynomial Rings

Sage implements sparse and dense polynomials over commutative and non-commutative rings. In the non-commutative case, the polynomial variable commutes with the elements of the base ring.

AUTHOR:

  • William Stein
  • Kiran Kedlaya (2006-02-13): added macaulay2 option
  • Martin Albrecht (2006-08-25): removed it again as it isn’t needed anymore
  • Simon King (2011-05): Dense and sparse polynomial rings must not be equal.
  • Simon King (2011-10): Choice of categories for polynomial rings.

EXAMPLES:

Creating a polynomial ring injects the variable into the interpreter namespace:

sage: z = QQ['z'].0
sage: (z^3 + z - 1)^3
z^9 + 3*z^7 - 3*z^6 + 3*z^5 - 6*z^4 + 4*z^3 - 3*z^2 + 3*z - 1

Saving and loading of polynomial rings works:

sage: loads(dumps(QQ['x'])) == QQ['x']
True
sage: k = PolynomialRing(QQ['x'],'y'); loads(dumps(k))==k
True
sage: k = PolynomialRing(ZZ,'y'); loads(dumps(k)) == k
True
sage: k = PolynomialRing(ZZ,'y', sparse=True); loads(dumps(k))
Sparse Univariate Polynomial Ring in y over Integer Ring

Rings with different variable names are not equal; in fact, by trac ticket #9944, poynomial rings are equal if and only if they are identic (which should be the case for all parent structures in Sage):

sage: QQ['y'] != QQ['x']
True
sage: QQ['y'] != QQ['z']
True

We create a polynomial ring over a quaternion algebra:

sage: A.<i,j,k> = QuaternionAlgebra(QQ, -1,-1)
sage: R.<w> = PolynomialRing(A,sparse=True)
sage: f = w^3 + (i+j)*w + 1
sage: f
w^3 + (i + j)*w + 1
sage: f^2
w^6 + (2*i + 2*j)*w^4 + 2*w^3 - 2*w^2 + (2*i + 2*j)*w + 1
sage: f = w + i ; g = w + j
sage: f * g
w^2 + (i + j)*w + k
sage: g * f
w^2 + (i + j)*w - k

Trac ticket #9944 introduced some changes related with coercion. Previously, a dense and a sparse polynomial ring with the same variable name over the same base ring evaluated equal, but of course they were not identical.Coercion maps are cached - but if a coercion to a dense ring is requested and a coercion to a sparse ring is returned instead (since the cache keys are equal!), all hell breaks loose.

Therefore, the coercion between rings of sparse and dense polynomials works as follows:

sage: R.<x> = PolynomialRing(QQ, sparse=True)
sage: S.<x> = QQ[]
sage: S == R
False
sage: S.has_coerce_map_from(R)
True
sage: R.has_coerce_map_from(S)
False
sage: (R.0+S.0).parent()
Univariate Polynomial Ring in x over Rational Field
sage: (S.0+R.0).parent()
Univariate Polynomial Ring in x over Rational Field

It may be that one has rings of dense or sparse polynomials over different base rings. In that situation, coercion works by means of the pushout() formalism:

sage: R.<x> = PolynomialRing(GF(5), sparse=True)
sage: S.<x> = PolynomialRing(ZZ)
sage: R.has_coerce_map_from(S)
False
sage: S.has_coerce_map_from(R)
False
sage: S.0 + R.0
2*x
sage: (S.0 + R.0).parent()
Univariate Polynomial Ring in x over Finite Field of size 5
sage: (S.0 + R.0).parent().is_sparse()
False

Similarly, there is a coercion from the (non-default) NTL implementation for univariate polynomials over the integers to the default FLINT implementation, but not vice versa:

sage: R.<x> = PolynomialRing(ZZ, implementation = 'NTL')
sage: S.<x> = PolynomialRing(ZZ, implementation = 'FLINT')
sage: (S.0+R.0).parent() is S
True
sage: (R.0+S.0).parent() is S
True

TESTS:

sage: K.<x>=FractionField(QQ['x'])
sage: V.<z> = K[]
sage: x+z
z + x

Check that trac ticket #5562 has been fixed:

sage: R.<u> = PolynomialRing(RDF, 1, 'u')
sage: v1 = vector([u])
sage: v2 = vector([CDF(2)])
sage: v1 * v2
2.0*u

These may change over time:

sage: x = var('x')
sage: type(ZZ['x'].0)
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>
sage: type(QQ['x'].0)
<type 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>
sage: type(RR['x'].0)
<type 'sage.rings.polynomial.polynomial_real_mpfr_dense.PolynomialRealDense'>
sage: type(Integers(4)['x'].0)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
sage: type(Integers(5*2^100)['x'].0)
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ'>
sage: type(CC['x'].0)
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>
sage: type(CC['t']['x'].0)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: type(NumberField(x^2+1,'I')['x'].0)
<class 'sage.rings.polynomial.polynomial_number_field.Polynomial_absolute_number_field_dense'>
sage: type(NumberField(x^2+1,'I')['x'])
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category'>
sage: type(NumberField([x^2-2,x^2-3],'a')['x'].0)
<class 'sage.rings.polynomial.polynomial_number_field.Polynomial_relative_number_field_dense'>
sage: type(NumberField([x^2-2,x^2-3],'a')[x])
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_commutative(base_ring, name=None, sparse=False, element_class=None, category=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_general, sage.rings.ring.CommutativeAlgebra

Univariate polynomial ring over a commutative ring.

quotient_by_principal_ideal(f, names=None)

Return the quotient of this polynomial ring by the principal ideal (generated by) \(f\).

INPUT:

  • f - either a polynomial in self, or a principal ideal of self.

EXAMPLES:

sage: R.<x> = QQ[]
sage: I = (x^2-1)*R
sage: R.quotient_by_principal_ideal(I)
Univariate Quotient Polynomial Ring in xbar over Rational Field with modulus x^2 - 1

The same example, using the polynomial instead of the ideal, and customizing the variable name:

sage: R.<x> = QQ[]
sage: R.quotient_by_principal_ideal(x^2-1, names=('foo',))
Univariate Quotient Polynomial Ring in foo over Rational Field with modulus x^2 - 1

TESTS:

Quotienting by the zero ideal returns self (trac ticket #5978):

sage: R = QQ['x']
sage: R.quotient_by_principal_ideal(R.zero_ideal()) is R
True
sage: R.quotient_by_principal_ideal(0) is R
True
weyl_algebra()

Return the Weyl algebra generated from self.

EXAMPLES:

sage: R = QQ['x']
sage: W = R.weyl_algebra(); W
Differential Weyl algebra of polynomials in x over Rational Field
sage: W.polynomial_ring() == R
True
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_finite_field(base_ring, name='x', element_class=None, implementation=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_field

Univariate polynomial ring over a finite field.

EXAMPLE:

sage: R = PolynomialRing(GF(27, 'a'), 'x')
sage: type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_finite_field_with_category'>
irreducible_element(n, algorithm=None)

Construct a monic irreducible polynomial of degree \(n\).

INPUT:

  • n – integer: degree of the polynomial to construct
  • algorithm – string: algorithm to use, or None
    • 'random': try random polynomials until an irreducible one is found.
    • 'first_lexicographic': try polynomials in lexicographic order until an irreducible one is found.

OUTPUT:

A monic irreducible polynomial of degree \(n\) in self.

EXAMPLES:

sage: GF(5^3, 'a')['x'].irreducible_element(2)
x^2 + (4*a^2 + a + 4)*x + 2*a^2 + 2
sage: GF(19)['x'].irreducible_element(21, algorithm="first_lexicographic")
x^21 + x + 5
sage: GF(5**2, 'a')['x'].irreducible_element(17, algorithm="first_lexicographic")
x^17 + a*x + 4*a + 3

AUTHORS:

  • Peter Bruin (June 2013)
  • Jean-Pierre Flori (May 2014)
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_n(base_ring, name=None, element_class=None, implementation=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_commutative

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_mod_n as PRing
sage: R = PRing(Zmod(15), 'x'); R
Univariate Polynomial Ring in x over Ring of integers modulo 15
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>

sage: R = PRing(Zmod(15), 'x', implementation='NTL'); R
Univariate Polynomial Ring in x over Ring of integers modulo 15 (using NTL)
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_zz'>

sage: R = PRing(Zmod(2**63*3), 'x', implementation='NTL'); R
Univariate Polynomial Ring in x over Ring of integers modulo 27670116110564327424 (using NTL)
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ'>

sage: R = PRing(Zmod(2**63*3), 'x', implementation='FLINT')
Traceback (most recent call last):
...
ValueError: FLINT does not support modulus 27670116110564327424

sage: R = PRing(Zmod(2**63*3), 'x'); R
Univariate Polynomial Ring in x over Ring of integers modulo 27670116110564327424 (using NTL)
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ'>
modulus()

EXAMPLES:

sage: R.<x> = Zmod(15)[]
sage: R.modulus()
15
residue_field(ideal, names=None)

Return the residue finite field at the given ideal.

EXAMPLES:

sage: R.<t> = GF(2)[]
sage: k.<a> = R.residue_field(t^3+t+1); k
Residue field in a of Principal ideal (t^3 + t + 1) of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL)
sage: k.list()
[0, a, a^2, a + 1, a^2 + a, a^2 + a + 1, a^2 + 1, 1]
sage: R.residue_field(t)
Residue field of Principal ideal (t) of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL)
sage: P = R.irreducible_element(8) * R
sage: P
Principal ideal (t^8 + t^4 + t^3 + t^2 + 1) of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL)
sage: k.<a> = R.residue_field(P); k
Residue field in a of Principal ideal (t^8 + t^4 + t^3 + t^2 + 1) of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL)
sage: k.cardinality()
256

Non-maximal ideals are not accepted:

sage: R.residue_field(t^2 + 1)
Traceback (most recent call last):
...
ArithmeticError: ideal is not maximal
sage: R.residue_field(0)
Traceback (most recent call last):
...
ArithmeticError: ideal is not maximal
sage: R.residue_field(1)
Traceback (most recent call last):
...
ArithmeticError: ideal is not maximal
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_p(base_ring, name='x', implementation=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_finite_field, sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_n, sage.rings.polynomial.polynomial_singular_interface.PolynomialRing_singular_repr

TESTS:

sage: P = GF(2)['x']; P
Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL)
sage: type(P.gen())
<type 'sage.rings.polynomial.polynomial_gf2x.Polynomial_GF2X'>

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_mod_p
sage: P = PolynomialRing_dense_mod_p(GF(5), 'x'); P
Univariate Polynomial Ring in x over Finite Field of size 5
sage: type(P.gen())
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>

sage: P = PolynomialRing_dense_mod_p(GF(5), 'x', implementation='NTL'); P
Univariate Polynomial Ring in x over Finite Field of size 5 (using NTL)
sage: type(P.gen())
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p'>

sage: P = PolynomialRing_dense_mod_p(GF(9223372036854775837), 'x')
sage: P
Univariate Polynomial Ring in x over Finite Field of size 9223372036854775837 (using NTL)
sage: type(P.gen())
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p'>
irreducible_element(n, algorithm=None)

Construct a monic irreducible polynomial of degree \(n\).

INPUT:

  • n – integer: the degree of the polynomial to construct

  • algorithm – string: algorithm to use, or None. Currently available options are:

    • 'adleman-lenstra': a variant of the Adleman–Lenstra

      algorithm as implemented in PARI.

    • 'conway': look up the Conway polynomial of degree \(n\) over the field of \(p\) elements in the database; raise a RuntimeError if it is not found.

    • 'ffprimroot': use the ffprimroot() function from PARI.

    • 'first_lexicographic': return the lexicographically smallest irreducible polynomial of degree \(n\).

    • 'minimal_weight': return an irreducible polynomial of degree \(n\) with minimal number of non-zero coefficients. Only implemented for \(p = 2\).

    • 'primitive': return a polynomial \(f\) such that a root of \(f\) generates the multiplicative group of the finite field extension defined by \(f\). This uses the Conway polynomial if possible, otherwise it uses ffprimroot.

    • 'random': try random polynomials until an irreducible one is found.

    If algorithm is None, use \(x - 1\) in degree 1. In degree > 1, the Conway polynomial is used if it is found in the database. Otherwise, the algorithm minimal_weight is used if \(p = 2\), and the algorithm adleman-lenstra if \(p > 2\).

OUTPUT:

A monic irreducible polynomial of degree \(n\) in self.

EXAMPLES:

sage: GF(5)['x'].irreducible_element(2)
x^2 + 4*x + 2
sage: GF(5)['x'].irreducible_element(2, algorithm="adleman-lenstra")
x^2 + x + 1
sage: GF(5)['x'].irreducible_element(2, algorithm="primitive")
x^2 + 4*x + 2
sage: GF(5)['x'].irreducible_element(32, algorithm="first_lexicographic")
x^32 + 2
sage: GF(5)['x'].irreducible_element(32, algorithm="conway")
Traceback (most recent call last):
...
RuntimeError: requested Conway polynomial not in database.
sage: GF(5)['x'].irreducible_element(32, algorithm="primitive")
x^32 + ...

In characteristic 2:

sage: GF(2)['x'].irreducible_element(33)
x^33 + x^13 + x^12 + x^11 + x^10 + x^8 + x^6 + x^3 + 1
sage: GF(2)['x'].irreducible_element(33, algorithm="minimal_weight")
x^33 + x^10 + 1

In degree 1:

sage: GF(97)['x'].irreducible_element(1)
x + 96
sage: GF(97)['x'].irreducible_element(1, algorithm="conway")
x + 92
sage: GF(97)['x'].irreducible_element(1, algorithm="adleman-lenstra")
x

AUTHORS:

  • Peter Bruin (June 2013)
  • Jeroen Demeyer (September 2014): add “ffprimroot” algorithm, see trac ticket #8373.
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_field_capped_relative(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_field_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_field_capped_relative as PRing
sage: R = PRing(Qp(13), name='t'); R
Univariate Polynomial Ring in t over 13-adic Field with capped relative precision 20
sage: type(R.gen())
<class 'sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense.Polynomial_padic_capped_relative_dense'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_field_generic(base_ring, name='x', sparse=False, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_field

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_field as PRing
sage: R = PRing(QQ, 'x'); R
Univariate Polynomial Ring in x over Rational Field
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>
sage: R = PRing(QQ, 'x', sparse=True); R
Sparse Univariate Polynomial Ring in x over Rational Field
sage: type(R.gen())
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_field'>
sage: R = PRing(CC, 'x'); R
Univariate Polynomial Ring in x over Complex Field with 53 bits of precision
sage: type(R.gen())
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>

#Demonstrate that Trac #8762 is fixed
sage: R.<x> = PolynomialRing(GF(next_prime(10^20)), sparse=True)
sage: x^(10^20) # this should be fast
x^100000000000000000000
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_field_lazy(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_field_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_field_lazy as PRing
sage: R = PRing(Qp(13, type='lazy'), name='t')
Traceback (most recent call last):
...
NotImplementedError: lazy p-adics need more work.  Sorry.

#sage: type(R.gen())
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_capped_absolute(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_ring_capped_absolute as PRing
sage: R = PRing(Zp(13, type='capped-abs'), name='t'); R
Univariate Polynomial Ring in t over 13-adic Ring with capped absolute precision 20
sage: type(R.gen())
<class 'sage.rings.polynomial.padics.polynomial_padic_flat.Polynomial_padic_flat'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_capped_relative(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_ring_capped_relative as PRing
sage: R = PRing(Zp(13), name='t'); R
Univariate Polynomial Ring in t over 13-adic Ring with capped relative precision 20
sage: type(R.gen())
<class 'sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense.Polynomial_padic_capped_relative_dense'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_fixed_mod(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_ring_fixed_mod as PRing
sage: R = PRing(Zp(13, type='fixed-mod'), name='t'); R
Univariate Polynomial Ring in t over 13-adic Ring of fixed modulus 13^20

sage: type(R.gen())
<class 'sage.rings.polynomial.padics.polynomial_padic_flat.Polynomial_padic_flat'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_generic(base_ring, name='x', sparse=False, implementation=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_integral_domain as PRing
sage: R = PRing(ZZ, 'x'); R
Univariate Polynomial Ring in x over Integer Ring
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>

sage: R = PRing(ZZ, 'x', implementation='NTL'); R
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl'>
class sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_lazy(base_ring, name=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_padic_ring_generic

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_dense_padic_ring_lazy as PRing
sage: R = PRing(Zp(13, type='lazy'), name='t')
Traceback (most recent call last):
...
NotImplementedError: lazy p-adics need more work.  Sorry.

#sage: type(R.gen())
class sage.rings.polynomial.polynomial_ring.PolynomialRing_field(base_ring, name='x', sparse=False, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain, sage.rings.polynomial.polynomial_singular_interface.PolynomialRing_singular_repr, sage.rings.ring.PrincipalIdealDomain

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_field as PRing
sage: R = PRing(QQ, 'x'); R
Univariate Polynomial Ring in x over Rational Field
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>
sage: R = PRing(QQ, 'x', sparse=True); R
Sparse Univariate Polynomial Ring in x over Rational Field
sage: type(R.gen())
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_field'>
sage: R = PRing(CC, 'x'); R
Univariate Polynomial Ring in x over Complex Field with 53 bits of precision
sage: type(R.gen())
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>

#Demonstrate that Trac #8762 is fixed
sage: R.<x> = PolynomialRing(GF(next_prime(10^20)), sparse=True)
sage: x^(10^20) # this should be fast
x^100000000000000000000
divided_difference(points, full_table=False)

Return the Newton divided-difference coefficients of the \(n\)-th Lagrange interpolation polynomial of points.

If points are \(n+1\) distinct points \((x_0, f(x_0)), (x_1, f(x_1)), \dots, (x_n, f(x_n))\), then \(P_n(x)\) is the \(n\)-th Lagrange interpolation polynomial of \(f(x)\) that passes through the points \((x_i, f(x_i))\). This method returns the coefficients \(F_{i,i}\) such that

\[P_n(x) = \sum_{i=0}^n F_{i,i} \prod_{j=0}^{i-1} (x - x_j)\]

INPUT:

  • points – a list of tuples \((x_0, f(x_0)), (x_1, f(x_1)), \dots, (x_n, f(x_n))\) where each \(x_i \neq x_j\) for \(i \neq j\)
  • full_table – (default: False) if True then return the full divided-difference table; if False then only return entries along the main diagonal. The entries along the main diagonal are the Newton divided-difference coefficients \(F_{i,i}\).

OUTPUT:

  • The Newton divided-difference coefficients of the \(n\)-th Lagrange interpolation polynomial that passes through the points in points.

EXAMPLES:

Only return the divided-difference coefficients \(F_{i,i}\). This example is taken from Example 1, p.121 of [BF05]:

sage: points = [(1.0, 0.7651977), (1.3, 0.6200860), (1.6, 0.4554022), (1.9, 0.2818186), (2.2, 0.1103623)]
sage: R = PolynomialRing(QQ, "x")
sage: R.divided_difference(points)

[0.765197700000000,
-0.483705666666666,
-0.108733888888889,
0.0658783950617283,
0.00182510288066044]

Now return the full divided-difference table:

sage: points = [(1.0, 0.7651977), (1.3, 0.6200860), (1.6, 0.4554022), (1.9, 0.2818186), (2.2, 0.1103623)]
sage: R = PolynomialRing(QQ, "x")
sage: R.divided_difference(points, full_table=True)

[[0.765197700000000],
[0.620086000000000, -0.483705666666666],
[0.455402200000000, -0.548946000000000, -0.108733888888889],
[0.281818600000000,
-0.578612000000000,
-0.0494433333333339,
0.0658783950617283],
[0.110362300000000,
-0.571520999999999,
0.0118183333333349,
0.0680685185185209,
0.00182510288066044]]

The following example is taken from Example 4.12, p.225 of [MF99]:

sage: points = [(1, -3), (2, 0), (3, 15), (4, 48), (5, 105), (6, 192)]
sage: R = PolynomialRing(RR, "x")
sage: R.divided_difference(points)
[-3, 3, 6, 1, 0, 0]
sage: R.divided_difference(points, full_table=True)

[[-3],
[0, 3],
[15, 15, 6],
[48, 33, 9, 1],
[105, 57, 12, 1, 0],
[192, 87, 15, 1, 0, 0]]

REFERENCES:

[MF99]J.H. Mathews and K.D. Fink. Numerical Methods Using MATLAB. 3rd edition, Prentice-Hall, 1999.
fraction_field()

Returns the fraction field of self.

EXAMPLES:

sage: R.<t> = GF(5)[]
sage: R.fraction_field()
Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5
lagrange_polynomial(points, algorithm='divided_difference', previous_row=None)

Return the Lagrange interpolation polynomial in self associated to the given list of points.

Given a list of points, i.e. tuples of elements of self‘s base ring, this function returns the interpolation polynomial in the Lagrange form.

INPUT:

  • points – a list of tuples representing points through which the polynomial returned by this function must pass.
  • algorithm – (default: 'divided_difference') the available values for this option are 'divided_difference' and neville.
    • If algorithm='divided_difference' then use the method of divided difference.
    • If algorithm='neville' then adapt Neville’s method as described on page 144 of [BF05] to recursively generate the Lagrange interpolation polynomial. Neville’s method generates a table of approximating polynomials, where the last row of that table contains the \(n\)-th Lagrange interpolation polynomial. The adaptation implemented by this method is to only generate the last row of this table, instead of the full table itself. Generating the full table can be memory inefficient.
  • previous_row – (default: None) This option is only relevant if used together with algorithm='neville'. If provided, this should be the last row of the table resulting from a previous use of Neville’s method. If such a row is passed in, then points should consist of both previous and new interpolating points. Neville’s method will then use that last row and the interpolating points to generate a new row which contains a better Lagrange interpolation polynomial.

EXAMPLES:

By default, we use the method of divided-difference:

sage: R = PolynomialRing(QQ, 'x')
sage: f = R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)]); f
-23/84*x^3 - 11/84*x^2 + 13/7*x + 1
sage: f(0)
1
sage: f(2)
2
sage: f(3)
-2
sage: f(-4)
9
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: f = R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)]); f
a^2*x^2 + a^2*x + a^2
sage: f(a^2+a)
a
sage: f(a)
1
sage: f(a^2)
a^2 + a + 1

Now use a memory efficient version of Neville’s method:

sage: R = PolynomialRing(QQ, 'x')
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], algorithm="neville")

[9,
-11/7*x + 19/7,
-17/42*x^2 - 83/42*x + 53/7,
-23/84*x^3 - 11/84*x^2 + 13/7*x + 1]
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)], algorithm="neville")
[a^2 + a + 1, x + a + 1, a^2*x^2 + a^2*x + a^2]

Repeated use of Neville’s method to get better Lagrange interpolation polynomials:

sage: R = PolynomialRing(QQ, 'x')
sage: p = R.lagrange_polynomial([(0,1),(2,2)], algorithm="neville")
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], algorithm="neville", previous_row=p)[-1]
-23/84*x^3 - 11/84*x^2 + 13/7*x + 1
sage: R = PolynomialRing(GF(2**3,'a'), 'x')
sage: a = R.base_ring().gen()
sage: p = R.lagrange_polynomial([(a^2+a,a),(a,1)], algorithm="neville")
sage: R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)], algorithm="neville", previous_row=p)[-1]
a^2*x^2 + a^2*x + a^2

TESTS:

The value for algorithm must be either 'divided_difference' (by default it is), or 'neville':

sage: R = PolynomialRing(QQ, "x")
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], algorithm="abc")
Traceback (most recent call last):
...
ValueError: algorithm must be one of 'divided_difference' or 'neville'
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], algorithm="divided difference")
Traceback (most recent call last):
...
ValueError: algorithm must be one of 'divided_difference' or 'neville'
sage: R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], algorithm="")
Traceback (most recent call last):
...
ValueError: algorithm must be one of 'divided_difference' or 'neville'

Make sure that ticket #10304 is fixed. The return value should always be an element of self in the case of divided_difference, or a list of elements of self in the case of neville.

sage: R = PolynomialRing(QQ, "x")
sage: R.lagrange_polynomial([]).parent() == R
True
sage: R.lagrange_polynomial([(2, 3)]).parent() == R
True
sage: row = R.lagrange_polynomial([], algorithm='neville')
sage: all(poly.parent() == R for poly in row)
True
sage: row = R.lagrange_polynomial([(2, 3)], algorithm='neville')
sage: all(poly.parent() == R for poly in row)
True

REFERENCES:

[BF05](1, 2) R.L. Burden and J.D. Faires. Numerical Analysis. Thomson Brooks/Cole, 8th edition, 2005.
class sage.rings.polynomial.polynomial_ring.PolynomialRing_general(base_ring, name=None, sparse=False, element_class=None, category=None)

Bases: sage.rings.ring.Algebra

Univariate polynomial ring over a ring.

base_extend(R)

Return the base extension of this polynomial ring to R.

EXAMPLES:

sage: R.<x> = RR[]; R
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: R.base_extend(CC)
Univariate Polynomial Ring in x over Complex Field with 53 bits of precision
sage: R.base_extend(QQ)
Traceback (most recent call last):
...
TypeError: no such base extension
sage: R.change_ring(QQ)
Univariate Polynomial Ring in x over Rational Field
change_ring(R)

Return the polynomial ring in the same variable as self over R.

EXAMPLES:

sage: R.<ZZZ> = RealIntervalField() []; R
Univariate Polynomial Ring in ZZZ over Real Interval Field with 53 bits of precision
sage: R.change_ring(GF(19^2,'b'))
Univariate Polynomial Ring in ZZZ over Finite Field in b of size 19^2
change_var(var)

Return the polynomial ring in variable var over the same base ring.

EXAMPLES:

sage: R.<x> = ZZ[]; R
Univariate Polynomial Ring in x over Integer Ring
sage: R.change_var('y')
Univariate Polynomial Ring in y over Integer Ring
characteristic()

Return the characteristic of this polynomial ring, which is the same as that of its base ring.

EXAMPLES:

sage: R.<ZZZ> = RealIntervalField() []; R
Univariate Polynomial Ring in ZZZ over Real Interval Field with 53 bits of precision
sage: R.characteristic()
0
sage: S = R.change_ring(GF(19^2,'b')); S
Univariate Polynomial Ring in ZZZ over Finite Field in b of size 19^2
sage: S.characteristic()
19
completion(p, prec=20, extras=None)

Return the completion of self with respect to the irreducible polynomial p. Currently only implemented for p=self.gen(), i.e. you can only complete R[x] with respect to x, the result being a ring of power series in x. The prec variable controls the precision used in the power series ring.

EXAMPLES:

sage: P.<x>=PolynomialRing(QQ)
sage: P
Univariate Polynomial Ring in x over Rational Field
sage: PP=P.completion(x)
sage: PP
Power Series Ring in x over Rational Field
sage: f=1-x
sage: PP(f)
1 - x
sage: 1/f
1/(-x + 1)
sage: 1/PP(f)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + O(x^20)
construction()

x.__init__(...) initializes x; see help(type(x)) for signature

cyclotomic_polynomial(n)

Return the nth cyclotomic polynomial as a polynomial in this polynomial ring. For details of the implementation, see the documentation for sage.rings.polynomial.cyclotomic.cyclotomic_coeffs().

EXAMPLES:

sage: R = ZZ['x']
sage: R.cyclotomic_polynomial(8)
x^4 + 1
sage: R.cyclotomic_polynomial(12)
x^4 - x^2 + 1
sage: S = PolynomialRing(FiniteField(7), 'x')
sage: S.cyclotomic_polynomial(12)
x^4 + 6*x^2 + 1
sage: S.cyclotomic_polynomial(1)
x + 6

TESTS:

Make sure it agrees with other systems for the trivial case:

sage: ZZ['x'].cyclotomic_polynomial(1)
x - 1
sage: gp('polcyclo(1)')
x - 1
extend_variables(added_names, order='degrevlex')

Returns a multivariate polynomial ring with the same base ring but with added_names as additional variables.

EXAMPLES:

sage: R.<x> = ZZ[]; R
Univariate Polynomial Ring in x over Integer Ring
sage: R.extend_variables('y, z')
Multivariate Polynomial Ring in x, y, z over Integer Ring
sage: R.extend_variables(('y', 'z'))
Multivariate Polynomial Ring in x, y, z over Integer Ring
gen(n=0)

Return the indeterminate generator of this polynomial ring.

EXAMPLES:

sage: R.<abc> = Integers(8)[]; R
Univariate Polynomial Ring in abc over Ring of integers modulo 8
sage: t = R.gen(); t
abc
sage: t.is_gen()
True

An identical generator is always returned.

sage: t is R.gen()
True
gens_dict()

Returns a dictionary whose keys are the variable names of this ring as strings and whose values are the corresponding generators.

EXAMPLES:

sage: R.<x> = RR[]
sage: R.gens_dict()
{'x': x}
is_exact()

x.__init__(...) initializes x; see help(type(x)) for signature

is_field(proof=True)

Return False, since polynomial rings are never fields.

EXAMPLES:

sage: R.<z> = Integers(2)[]; R
Univariate Polynomial Ring in z over Ring of integers modulo 2 (using NTL)
sage: R.is_field()
False
is_finite()

Return False since polynomial rings are not finite (unless the base ring is 0.)

EXAMPLES:

sage: R = Integers(1)['x']
sage: R.is_finite()
True
sage: R = GF(7)['x']
sage: R.is_finite()
False
sage: R['x']['y'].is_finite()
False
is_integral_domain(proof=True)

EXAMPLES:

sage: ZZ['x'].is_integral_domain()
True
sage: Integers(8)['x'].is_integral_domain()
False
is_noetherian()

x.__init__(...) initializes x; see help(type(x)) for signature

is_sparse()

Return true if elements of this polynomial ring have a sparse representation.

EXAMPLES:

sage: R.<z> = Integers(8)[]; R
Univariate Polynomial Ring in z over Ring of integers modulo 8
sage: R.is_sparse()
False
sage: R.<W> = PolynomialRing(QQ, sparse=True); R
Sparse Univariate Polynomial Ring in W over Rational Field
sage: R.is_sparse()
True
karatsuba_threshold()

Return the Karatsuba threshold used for this ring by the method _mul_karatsuba to fall back to the schoolbook algorithm.

EXAMPLES:

sage: K = QQ['x']
sage: K.karatsuba_threshold()
8
sage: K = QQ['x']['y']
sage: K.karatsuba_threshold()
0
krull_dimension()

Return the Krull dimension of this polynomial ring, which is one more than the Krull dimension of the base ring.

EXAMPLES:

sage: R.<x> = QQ[]
sage: R.krull_dimension()
1
sage: R.<z> = GF(9,'a')[]; R
Univariate Polynomial Ring in z over Finite Field in a of size 3^2
sage: R.krull_dimension()
1
sage: S.<t> = R[]
sage: S.krull_dimension()
2
sage: for n in range(10):
...    S = PolynomialRing(S,'w')
sage: S.krull_dimension()
12
monics(of_degree=None, max_degree=None)

Return an iterator over the monic polynomials of specified degree.

INPUT: Pass exactly one of:

  • max_degree - an int; the iterator will generate all monic polynomials which have degree less than or equal to max_degree
  • of_degree - an int; the iterator will generate all monic polynomials which have degree of_degree

OUTPUT: an iterator

EXAMPLES:

sage: P = PolynomialRing(GF(4,'a'),'y')
sage: for p in P.monics( of_degree = 2 ): print p
y^2
y^2 + a
y^2 + a + 1
y^2 + 1
y^2 + a*y
y^2 + a*y + a
y^2 + a*y + a + 1
y^2 + a*y + 1
y^2 + (a + 1)*y
y^2 + (a + 1)*y + a
y^2 + (a + 1)*y + a + 1
y^2 + (a + 1)*y + 1
y^2 + y
y^2 + y + a
y^2 + y + a + 1
y^2 + y + 1
sage: for p in P.monics( max_degree = 1 ): print p
1
y
y + a
y + a + 1
y + 1
sage: for p in P.monics( max_degree = 1, of_degree = 3 ): print p
Traceback (most recent call last):
...
ValueError: you should pass exactly one of of_degree and max_degree

AUTHORS:

  • Joel B. Mohler
ngens()

Return the number of generators of this polynomial ring, which is 1 since it is a univariate polynomial ring.

EXAMPLES:

sage: R.<z> = Integers(8)[]; R
Univariate Polynomial Ring in z over Ring of integers modulo 8
sage: R.ngens()
1
parameter()

Return the generator of this polynomial ring.

This is the same as self.gen().

polynomials(of_degree=None, max_degree=None)

Return an iterator over the polynomials of specified degree.

INPUT: Pass exactly one of:

  • max_degree - an int; the iterator will generate all polynomials which have degree less than or equal to max_degree
  • of_degree - an int; the iterator will generate all polynomials which have degree of_degree

OUTPUT: an iterator

EXAMPLES:

sage: P = PolynomialRing(GF(3),'y')
sage: for p in P.polynomials( of_degree = 2 ): print p
y^2
y^2 + 1
y^2 + 2
y^2 + y
y^2 + y + 1
y^2 + y + 2
y^2 + 2*y
y^2 + 2*y + 1
y^2 + 2*y + 2
2*y^2
2*y^2 + 1
2*y^2 + 2
2*y^2 + y
2*y^2 + y + 1
2*y^2 + y + 2
2*y^2 + 2*y
2*y^2 + 2*y + 1
2*y^2 + 2*y + 2
sage: for p in P.polynomials( max_degree = 1 ): print p
0
1
2
y
y + 1
y + 2
2*y
2*y + 1
2*y + 2
sage: for p in P.polynomials( max_degree = 1, of_degree = 3 ): print p
Traceback (most recent call last):
...
ValueError: you should pass exactly one of of_degree and max_degree

AUTHORS:

  • Joel B. Mohler
random_element(degree=(-1, 2), *args, **kwds)

Return a random polynomial of given degree or with given degree bounds.

INPUT:

  • degree - optional integer for fixing the degree or or a tuple of minimum and maximum degrees. By default set to (-1,2).
  • *args, **kwds - Passed on to the random_element method for the base ring

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R.random_element(10, 5,10)
9*x^10 + 8*x^9 + 6*x^8 + 8*x^7 + 8*x^6 + 9*x^5 + 8*x^4 + 8*x^3 + 6*x^2 + 8*x + 8
sage: R.random_element(6)
x^6 - 3*x^5 - x^4 + x^3 - x^2 + x + 1
sage: R.random_element(6)
-2*x^6 - 2*x^5 + 2*x^4 - 3*x^3 + 1
sage: R.random_element(6)
-x^6 + x^5 - x^4 + 4*x^3 - x^2 + x

If a tuple of two integers is given for the degree argument, a polynomial of degree in between the bound is given:

sage: R.random_element(degree=(0,8))
x^8 + 4*x^7 + 2*x^6 - x^4 + 4*x^3 - 5*x^2 + x + 14
sage: R.random_element(degree=(0,8))
-5*x^7 + x^6 - 3*x^5 + 4*x^4 - x^2 - 2*x + 1

Note that the zero polynomial has degree -1, so if you want to consider it set the minimum degree to -1:

sage: any(R.random_element(degree=(-1,2),x=-1,y=1) == R.zero() for _ in xrange(100))
True

TESTS:

sage: R.random_element(degree=[5])
Traceback (most recent call last):
...
ValueError: degree argument must be an integer or a tuple of 2 integers (min_degree, max_degree)

sage: R.random_element(degree=(5,4))
Traceback (most recent call last):
...
ValueError: minimum degree must be less or equal than maximum degree

Check that trac ticket #16682 is fixed:

sage: R = PolynomialRing(GF(2), 'z')
sage: for _ in xrange(100):
....:     d = randint(-1,20)
....:     P = R.random_element(degree=d)
....:     assert P.degree() == d, "problem with {} which has not degree {}".format(P,d)

sage: R.random_element(degree=-2)
Traceback (most recent call last):
...
ValueError: degree should be an integer greater or equal than -1
set_karatsuba_threshold(Karatsuba_threshold)

Changes the default threshold for this ring in the method _mul_karatsuba to fall back to the schoolbook algorithm.

Warning

This method may have a negative performance impact in polynomial arithmetic. So use it at your own risk.

EXAMPLES:

sage: K = QQ['x']
sage: K.karatsuba_threshold()
8
sage: K.set_karatsuba_threshold(0)
sage: K.karatsuba_threshold()
0
some_elements()

Return a list of polynomials.

This is typically used for running generic tests.

EXAMPLES:

sage: R.<x> = QQ[]
sage: R.some_elements()
[x, 0, 1, 1/2, x^2 + 2*x + 1, x^3, x^2 - 1, x^2 + 1, 2*x^2 + 2]
variable_names_recursive(depth=+Infinity)

Returns the list of variable names of this and its base rings, as if it were a single multi-variate polynomial.

EXAMPLES:

sage: R = QQ['x']['y']['z']
sage: R.variable_names_recursive()
('x', 'y', 'z')
sage: R.variable_names_recursive(2)
('y', 'z')
class sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain(base_ring, name='x', sparse=False, implementation=None, element_class=None)

Bases: sage.rings.polynomial.polynomial_ring.PolynomialRing_commutative, sage.rings.ring.IntegralDomain

TESTS:

sage: from sage.rings.polynomial.polynomial_ring import PolynomialRing_integral_domain as PRing
sage: R = PRing(ZZ, 'x'); R
Univariate Polynomial Ring in x over Integer Ring
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>

sage: R = PRing(ZZ, 'x', implementation='NTL'); R
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: type(R.gen())
<type 'sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl'>
sage.rings.polynomial.polynomial_ring.is_PolynomialRing(x)

Return True if x is a univariate polynomial ring (and not a sparse multivariate polynomial ring in one variable).

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
sage: from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
sage: is_PolynomialRing(2)
False

This polynomial ring is not univariate.

sage: is_PolynomialRing(ZZ['x,y,z'])
False
sage: is_MPolynomialRing(ZZ['x,y,z'])
True
sage: is_PolynomialRing(ZZ['w'])
True

Univariate means not only in one variable, but is a specific data type. There is a multivariate (sparse) polynomial ring data type, which supports a single variable as a special case.

sage: is_PolynomialRing(PolynomialRing(ZZ,1,'w'))
False
sage: R = PolynomialRing(ZZ,1,'w'); R
Multivariate Polynomial Ring in w over Integer Ring
sage: is_PolynomialRing(R)
False
sage: type(R)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
sage.rings.polynomial.polynomial_ring.polygen(ring_or_element, name='x')

Return a polynomial indeterminate.

INPUT:

  • polygen(base_ring, name=”x”)
  • polygen(ring_element, name=”x”)

If the first input is a ring, return a polynomial generator over that ring. If it is a ring element, return a polynomial generator over the parent of the element.

EXAMPLES:

sage: z = polygen(QQ,'z')
sage: z^3 + z +1
z^3 + z + 1
sage: parent(z)
Univariate Polynomial Ring in z over Rational Field

Note

If you give a list or comma separated string to polygen, you’ll get a tuple of indeterminates, exactly as if you called polygens.

sage.rings.polynomial.polynomial_ring.polygens(base_ring, names='x')

Return indeterminates over the given base ring with the given names.

EXAMPLES:

sage: x,y,z = polygens(QQ,'x,y,z')
sage: (x+y+z)^2
x^2 + 2*x*y + y^2 + 2*x*z + 2*y*z + z^2
sage: parent(x)
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: t = polygens(QQ,['x','yz','abc'])
sage: t
(x, yz, abc)

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