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

sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing(base_ring, *args, **kwds)

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

There are many ways to specify the variables for the polynomial ring:

1. PolynomialRing(base_ring, name, ...)
2. PolynomialRing(base_ring, names, ...)
3. PolynomialRing(base_ring, n, names, ...)
4. PolynomialRing(base_ring, n, ..., var_array=var_array, ...)

The ... at the end of these commands stands for additional keywords, like sparse or order.

INPUT:

• base_ring – a ring
• n – an integer
• name – a string
• names – a list or tuple of names (strings), or a comma separated string
• var_array – a list or tuple of names, or a comma separated string
• sparse – bool: whether or not elements are sparse. The default is a dense representation (sparse=False) for univariate rings and a sparse representation (sparse=True) for multivariate rings.
• 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'). For many base rings, the "singular" implementation is available. One can always specify implementation="generic" for a generic Sage implementation which does not use any specialized library.

Note

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:

1. PolynomialRing(base_ring, name, …)

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


The generic implementation uses neither NTL nor FLINT:

sage: Zx = PolynomialRing(ZZ, 'x', implementation='generic'); Zx
Univariate Polynomial Ring in x over Integer Ring
sage: Zx.element_class
<... 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>


2. PolynomialRing(base_ring, names, …)

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
True
sage: 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


The Singular implementation always returns a multivariate ring, even for 1 variable:

sage: PolynomialRing(QQ, "x", implementation="singular")
Multivariate Polynomial Ring in x over Rational Field
sage: P.<x> = PolynomialRing(QQ, implementation="singular"); P
Multivariate Polynomial Ring in x over Rational Field


3. PolynomialRing(base_ring, n, names, …) (where the arguments n and names may be reversed)

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(QQ, 2, 'alpha0')
Multivariate Polynomial Ring in alpha00, alpha01 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 primes less than 100:

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


4. PolynomialRing(base_ring, n, …, var_array=var_array, …)

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


It is possible to create higher-dimensional arrays:

sage: PolynomialRing(ZZ, 2, 3, var_array=('p', 'q'))
Multivariate Polynomial Ring in p00, q00, p01, q01, p02, q02, p10, q10, p11, q11, p12, q12 over Integer Ring
sage: PolynomialRing(ZZ, 2, 3, 4, var_array='m')
Multivariate Polynomial Ring in m000, m001, m002, m003, m010, m011, m012, m013, m020, m021, m022, m023, m100, m101, m102, m103, m110, m111, m112, m113, m120, m121, m122, m123 over Integer Ring


The array is always at least 2-dimensional. So, if var_array is a single string and only a single number $$n$$ is given, 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


Square brackets notation

You can alternatively create a polynomial ring over a ring $$R$$ with square brackets:

sage: RR["x"]
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: RR["x,y"]
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision
sage: P.<x,y> = RR[]; P
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision


This notation does not allow to set any of the optional arguments.

Changing variable names

Consider

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

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

Choose an appropriate category for a polynomial ring.

It is assumed that the corresponding base ring is nonzero.

INPUT:

• base_ring_category – The category of ring over which the polynomial ring shall be defined
• n_variables – number of variables

EXAMPLES:

sage: from sage.rings.polynomial.polynomial_ring_constructor import polynomial_default_category
sage: polynomial_default_category(Rings(),1) is Algebras(Rings()).Infinite()
True
sage: polynomial_default_category(Rings().Commutative(),1) is Algebras(Rings().Commutative()).Commutative().Infinite()
True
sage: polynomial_default_category(Fields(),1) is EuclideanDomains() & Algebras(Fields()).Infinite()
True
sage: polynomial_default_category(Fields(),2) is UniqueFactorizationDomains() & CommutativeAlgebras(Fields()).Infinite()
True

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

sage.rings.polynomial.polynomial_ring_constructor.unpickle_PolynomialRing(base_ring, arg1=None, arg2=None, sparse=False)

Custom unpickling function for polynomial rings.

This has the same positional arguments as the old PolynomialRing constructor before trac ticket #23338.