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 stringsorder
– 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:
PolynomialRing(base_ring, name, ...)
PolynomialRing(base_ring, names, ...)
PolynomialRing(base_ring, n, names, ...)
PolynomialRing(base_ring, n, ..., var_array=var_array, ...)
The
...
at the end of these commands stands for additional keywords, likesparse
ororder
.INPUT:
base_ring
– a ringn
– an integername
– a stringnames
– a list or tuple of names (strings), or a comma separated stringvar_array
– a list or tuple of names, or a comma separated stringsparse
– 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 orTermOrder
object, e.g.,'degrevlex'
(default) – degree reverse lexicographic'lex'
– lexicographic'deglex'
– degree lexicographicTermOrder('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 specifyimplementation="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
andnames
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 definedn_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.