# Constructors for polynomial rings¶

This module provides the function PolynomialRing(), which constructs rings of univariate and multivariate polynomials, and implements caching to prevent the same ring being created in memory multiple times (which is wasteful and breaks the general assumption in Sage that parents are unique).

There is also a function BooleanPolynomialRing_constructor(), used for constructing Boolean polynomial rings, which are not technically polynomial rings but rather quotients of them (see module sage.rings.polynomial.pbori for more details).

sage.rings.polynomial.polynomial_ring_constructor.BooleanPolynomialRing_constructor(n=None, names=None, order='lex')

Construct a boolean polynomial ring with the following parameters:

INPUT:

• n – number of variables (an integer > 1)
• names – names of ring variables, may be a string or list/tuple of strings
• order – term order (default: lex)

EXAMPLES:

sage: R.<x, y, z> = BooleanPolynomialRing() # indirect doctest
sage: R
Boolean PolynomialRing in x, y, z

sage: p = x*y + x*z + y*z
sage: x*p
x*y*z + x*y + x*z

sage: R.term_order()
Lexicographic term order

sage: R = BooleanPolynomialRing(5,'x',order='deglex(3),deglex(2)')
sage: R.term_order()
Block term order with blocks:
(Degree lexicographic term order of length 3,
Degree lexicographic term order of length 2)

sage: R = BooleanPolynomialRing(3,'x',order='degneglex')
sage: R.term_order()
Degree negative lexicographic term order

sage: BooleanPolynomialRing(names=('x','y'))
Boolean PolynomialRing in x, y

sage: BooleanPolynomialRing(names='x,y')
Boolean PolynomialRing in x, y


TESTS:

sage: P.<x,y> = BooleanPolynomialRing(2,order='deglex')
sage: x > y
True

sage: P.<x0, x1, x2, x3> = BooleanPolynomialRing(4,order='deglex(2),deglex(2)')
sage: x0 > x1
True
sage: x2 > x3
True

sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing(base_ring, arg1=None, arg2=None, sparse=False, order='degrevlex', names=None, name=None, var_array=None, implementation=None)

Return the globally unique univariate or multivariate polynomial ring with given properties and variable name or names.

There are five ways to call the polynomial ring constructor:

1. PolynomialRing(base_ring, name,    sparse=False)
2. PolynomialRing(base_ring, names,   order='degrevlex')
3. PolynomialRing(base_ring, name, n, order='degrevlex')
4. PolynomialRing(base_ring, n, name, order='degrevlex')
5. PolynomialRing(base_ring, n, var_array=var_array, order='degrevlex')

The optional arguments sparse and order must be explicitly named, and the other arguments must be given positionally.

INPUT:

• base_ring – a ring
• name – a string
• names – a list or tuple of names, or a comma separated string
• var_array – a list or tuple of names, or a comma separated string
• n – an integer
• sparse – bool (default: False), whether or not elements are sparse
• order – string or TermOrder object, e.g.,
• 'degrevlex' (default) – degree reverse lexicographic
• 'lex' – lexicographic
• 'deglex' – degree lexicographic
• TermOrder('deglex',3) + TermOrder('deglex',3) – block ordering
• implementation – string or None; selects an implementation in cases where Sage includes multiple choices (currently $$\ZZ[x]$$ can be implemented with ‘NTL’ or ‘FLINT’; default is ‘FLINT’)

Note

The following rules were introduced in trac ticket #9944, in order to preserve the “unique parent assumption” in Sage (i.e., if two parents evaluate equal then they should actually be identical).

• In the multivariate case, a dense representation is not supported. Hence, the argument sparse=False is silently ignored in that case.
• If the given implementation does not exist for rings with the given number of generators and the given sparsity, then an error results.

OUTPUT:

PolynomialRing(base_ring, name, sparse=False) returns a univariate polynomial ring; also, PolynomialRing(base_ring, names, sparse=False) yields a univariate polynomial ring, if names is a list or tuple providing exactly one name. All other input formats return a multivariate polynomial ring.

UNIQUENESS and IMMUTABILITY: In Sage there is exactly one single-variate polynomial ring over each base ring in each choice of variable, sparseness, and implementation. There is also exactly one multivariate polynomial ring over each base ring for each choice of names of variables and term order. The names of the generators can only be temporarily changed after the ring has been created. Do this using the localvars context:

EXAMPLES of VARIABLE NAME CONTEXT:

sage: R.<x,y> = PolynomialRing(QQ,2); R
Multivariate Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^2


You can’t just globally change the names of those variables. This is because objects all over Sage could have pointers to that polynomial ring.

sage: R._assign_names(['z','w'])
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.


However, you can very easily change the names within a with block:

sage: with localvars(R, ['z','w']):
...     print f
...
z^2 - 2*w^2


After the with block the names revert to what they were before.

sage: print f
x^2 - 2*y^2


SQUARE BRACKETS NOTATION: You can alternatively create a single or multivariate polynomial ring over a ring $$R$$ by writing R['varname'] or R['var1,var2,var3,...']. This square brackets notation doesn’t allow for setting any of the optional arguments.

EXAMPLES:

1. PolynomialRing(base_ring, name, sparse=False)

sage: PolynomialRing(QQ, 'w')
Univariate Polynomial Ring in w over Rational Field


Use the diamond brackets notation to make the variable ready for use after you define the ring:

sage: R.<w> = PolynomialRing(QQ)
sage: (1 + w)^3
w^3 + 3*w^2 + 3*w + 1


You must specify a name:

sage: PolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: You must specify the names of the variables.

sage: R.<abc> = PolynomialRing(QQ, sparse=True); R
Sparse Univariate Polynomial Ring in abc over Rational Field

sage: R.<w> = PolynomialRing(PolynomialRing(GF(7),'k')); R
Univariate Polynomial Ring in w over Univariate Polynomial Ring in k over Finite Field of size 7


The square bracket notation:

sage: R.<y> = QQ['y']; R
Univariate Polynomial Ring in y over Rational Field
sage: y^2 + y
y^2 + y


In fact, since the diamond brackets on the left determine the variable name, you can omit the variable from the square brackets:

sage: R.<zz> = QQ[]; R
Univariate Polynomial Ring in zz over Rational Field
sage: (zz + 1)^2
zz^2 + 2*zz + 1


This is exactly the same ring as what PolynomialRing returns:

sage: R is PolynomialRing(QQ,'zz')
True


However, rings with different variables are different:

sage: QQ['x'] == QQ['y']
False


Sage has two implementations of univariate polynomials over the integers, one based on NTL and one based on FLINT. The default is FLINT. Note that FLINT uses a “more dense” representation for its polynomials than NTL, so in particular, creating a polynomial like 2^1000000 * x^1000000 in FLINT may be unwise.

sage: ZxNTL = PolynomialRing(ZZ, 'x', implementation='NTL'); ZxNTL
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: ZxFLINT = PolynomialRing(ZZ, 'x', implementation='FLINT'); ZxFLINT
Univariate Polynomial Ring in x over Integer Ring
sage: ZxFLINT is ZZ['x']
True
sage: ZxFLINT is PolynomialRing(ZZ, 'x')
True
sage: xNTL = ZxNTL.gen()
sage: xFLINT = ZxFLINT.gen()
sage: xNTL.parent()
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: xFLINT.parent()
Univariate Polynomial Ring in x over Integer Ring


There is a coercion from the non-default to the default implementation, so the values can be mixed in a single expression:

sage: (xNTL + xFLINT^2)
x^2 + x


The result of such an expression will use the default, i.e., the FLINT implementation:

sage: (xNTL + xFLINT^2).parent()
Univariate Polynomial Ring in x over Integer Ring

2. PolynomialRing(base_ring, names,   order='degrevlex')

sage: R = PolynomialRing(QQ, 'a,b,c'); R
Multivariate Polynomial Ring in a, b, c over Rational Field

sage: S = PolynomialRing(QQ, ['a','b','c']); S
Multivariate Polynomial Ring in a, b, c over Rational Field

sage: T = PolynomialRing(QQ, ('a','b','c')); T
Multivariate Polynomial Ring in a, b, c over Rational Field


All three rings are identical.

sage: (R is S) and  (S is T)
True


There is a unique polynomial ring with each term order:

sage: R = PolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: S = PolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: S is PolynomialRing(QQ, 'x,y,z', order='invlex')
True
sage: R == S
False


Note that a univariate polynomial ring is returned, if the list of names is of length one. If it is of length zero, a multivariate polynomial ring with no variables is returned.

sage: PolynomialRing(QQ,["x"])
Univariate Polynomial Ring in x over Rational Field
sage: PolynomialRing(QQ,[])
Multivariate Polynomial Ring in no variables over Rational Field

3. PolynomialRing(base_ring, name, n, order='degrevlex')

If you specify a single name as a string and a number of variables, then variables labeled with numbers are created.

sage: PolynomialRing(QQ, 'x', 10)
Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field

sage: PolynomialRing(GF(7), 'y', 5)
Multivariate Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7

sage: PolynomialRing(QQ, 'y', 3, sparse=True)
Multivariate Polynomial Ring in y0, y1, y2 over Rational Field


Note that a multivariate polynomial ring is returned when an explicit number is given.

sage: PolynomialRing(QQ,"x",1)
Multivariate Polynomial Ring in x over Rational Field
sage: PolynomialRing(QQ,"x",0)
Multivariate Polynomial Ring in no variables over Rational Field


It is easy in Python to create fairly arbitrary variable names. For example, here is a ring with generators labeled by the first 100 primes:

sage: R = PolynomialRing(ZZ, ['x%s'%p for p in primes(100)]); R
Multivariate Polynomial Ring in x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 over Integer Ring


By calling the inject_variables() method, all those variable names are available for interactive use:

sage: R.inject_variables()
Defining x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97
sage: (x2 + x41 + x71)^2
x2^2 + 2*x2*x41 + x41^2 + 2*x2*x71 + 2*x41*x71 + x71^2

1. PolynomialRing(base_ring, n, m, var_array=var_array, order='degrevlex')

This creates an array of variables where each variables begins with an entry in var_array and is indexed from 0 to n-1.

sage: PolynomialRing(ZZ, 3, var_array=[‘x’,’y’]) Multivariate Polynomial Ring in x0, y0, x1, y1, x2, y2 over Integer Ring sage: PolynomialRing(ZZ, 3, var_array=’a,b’) Multivariate Polynomial Ring in a0, b0, a1, b1, a2, b2 over Integer Ring

If var_array is a single string, this creates an $$m \times n$$ array of variables:

sage: PolynomialRing(ZZ, 2, 3, var_array='m')
Multivariate Polynomial Ring in m00, m01, m02, m10, m11, m12 over Integer Ring


If var_array is a single string and $$m$$ is not specified, this creates an $$n \times n$$ array of variables:

sage: PolynomialRing(ZZ, 2, var_array='m')
Multivariate Polynomial Ring in m00, m01, m10, m11 over Integer Ring


TESTS:

We test here some changes introduced in trac ticket #9944.

If there is no dense implementation for the given number of variables, then requesting a dense ring results yields the corresponding sparse ring:

sage: R.<x,y> = QQ[]
sage: S.<x,y> = PolynomialRing(QQ, sparse=False)
sage: R is S
True


If the requested implementation is not known or not supported for the given number of variables and the given sparsity, then an error results:

sage: R.<x> = PolynomialRing(ZZ, implementation='Foo')
Traceback (most recent call last):
...
ValueError: Unknown implementation Foo for ZZ[x]
sage: R.<x,y> = PolynomialRing(ZZ, implementation='FLINT')
Traceback (most recent call last):
...
ValueError: The FLINT implementation is not known for multivariate polynomial rings


The following corner case used to result in a warning message from libSingular, and the generators of the resulting polynomial ring were not zero:

sage: R = Integers(1)['x','y']
sage: R.0 == 0
True


We verify that trac ticket #13187 is fixed:

sage: var('t')
t
sage: PolynomialRing(ZZ, name=t) == PolynomialRing(ZZ, name='t')
True


We verify that polynomials with interval coefficients from trac ticket #7712 and trac ticket #13760 are fixed:

sage: P.<y,z> = PolynomialRing(RealIntervalField(2))
sage: Q.<x> = PolynomialRing(P)
sage: C = (y-x)^3
sage: C(y/2)
1.?*y^3
sage: R.<x,y> = PolynomialRing(RIF,2)
sage: RIF(-2,1)*x
0.?e1*x

sage.rings.polynomial.polynomial_ring_constructor.polynomial_default_category(base_ring_category, multivariate)

Choose an appropriate category for a polynomial ring.

INPUT:

• base_ring_category: The category of ring over which the polynomial ring shall be defined.
• multivariate: Will the polynomial ring be multivariate?

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_ring_constructor import polynomial_default_category
sage: polynomial_default_category(Rings(), False) is Algebras(Rings())
True
sage: polynomial_default_category(Rings().Commutative(),False) is Algebras(Rings().Commutative()).Commutative()
True
sage: polynomial_default_category(Fields(),False) is EuclideanDomains() & Algebras(Fields())
True
sage: polynomial_default_category(Fields(),True) is UniqueFactorizationDomains() & CommutativeAlgebras(Fields())
True

sage: QQ['t'].category() is EuclideanDomains() & CommutativeAlgebras(QQ.category())
True
sage: QQ['s','t'].category() is UniqueFactorizationDomains() & CommutativeAlgebras(QQ.category())
True
sage: QQ['s']['t'].category() is UniqueFactorizationDomains() & CommutativeAlgebras(QQ['s'].category())
True


Polynomial Rings

#### Next topic

Univariate Polynomials and Polynomial Rings