Symmetric Ideals of Infinite Polynomial Rings

This module provides an implementation of ideals of polynomial rings in a countably infinite number of variables that are invariant under variable permutation. Such ideals are called ‘Symmetric Ideals’ in the rest of this document. Our implementation is based on the theory of M. Aschenbrenner and C. Hillar.

AUTHORS:

class sage.rings.polynomial.symmetric_ideal.SymmetricIdeal(ring, gens, coerce=True)

Ideal in an Infinite Polynomial Ring, invariant under permutation of variable indices

THEORY:

An Infinite Polynomial Ring with finitely many generators x_\ast, y_\ast, ... over a field F is a free commutative F-algebra generated by infinitely many ‘variables’ x_0, x_1, x_2,..., y_0, y_1, y_2,.... We refer to the natural number n as the index of the variable x_n. See more detailed description at infinite_polynomial_ring

Infinite Polynomial Rings are equipped with a permutation action by permuting positive variable indices, i.e., x_n^P = x_{P(n)}, y_n^P=y_{P(n)}, ... for any permutation P. Note that the variables x_0, y_0, ... of index zero are invariant under that action.

A Symmetric Ideal is an ideal in an infinite polynomial ring X that is invariant under the permutation action. In other words, if \mathfrak S_\infty denotes the symmetric group of 1,2,..., then a Symmetric Ideal is a right X[\mathfrak S_\infty]-submodule of X.

It is known by work of Aschenbrenner and Hillar [AB2007] that an Infinite Polynomial Ring X with a single generator x_\ast is Noetherian, in the sense that any Symmetric Ideal I\subset X is finitely generated modulo addition, multiplication by elements of X, and permutation of variable indices (hence, it is a finitely generated right X[\mathfrak S_\infty]-module).

Moreover, if X is equipped with a lexicographic monomial ordering with x_1 < x_2 < x_3 ... then there is an algorithm of Buchberger type that computes a Groebner basis G for I that allows for computation of a unique normal form, that is zero precisely for the elements of I – see [AB2008]. See groebner_basis() for more details.

Our implementation allows more than one generator and also provides degree lexicographic and degree reverse lexicographic monomial orderings – we do, however, not guarantee termination of the Buchberger algorithm in these cases.

[AB2007]M. Aschenbrenner, C. Hillar, Finite generation of symmetric ideals. Trans. Amer. Math. Soc. 359 (2007), no. 11, 5171–5192.
[AB2008](1, 2, 3) M. Aschenbrenner, C. Hillar, An Algorithm for Finding Symmetric Groebner Bases in Infinite Dimensional Rings.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]^2+y[2]^2,x[1]*x[2]*y[3]+x[1]*y[4])
sage: I == loads(dumps(I))
True
sage: latex(I)
\left(y_{2}^{2} + x_{1}^{2}, y_{4} x_{1} + y_{3} x_{2} x_{1}\right)\Bold{Q}[x_{\ast}, y_{\ast}][\mathfrak{S}_{\infty}]

The default ordering is lexicographic. We now compute a Groebner basis:

sage: J=I.groebner_basis()
sage: J
[x1^4 + x1^3, x2*x1^2 - x1^3, x2^2 - x1^2, y1*x1^3 + y1*x1^2, y1*x2 + y1*x1^2, y1^2 + x1^2, y2*x1 + y1*x1^2]

Ideal membership in I can now be tested by commuting symmetric reduction modulo J:

sage: I.reduce(J)
Symmetric Ideal (0, 0) of Infinite polynomial ring in x, y over Rational Field

Note that the Groebner basis is not point-wise invariant under permutation. However, any element of J has symmetric reduction zero even after applying a permutation:

sage: P=Permutation([1, 4, 3, 2])
sage: J[2]
x2^2 - x1^2
sage: J[2]^P
x4^2 - x1^2
sage: J.__contains__(J[2]^P)
False
sage: [[(p^P).reduce(J) for p in J] for P in Permutations(4)]
[[0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 0, 0, 0]]

Since I is not a Groebner basis, it is no surprise that it can not detect ideal membership:

sage: [p.reduce(I) for p in J]
[x1^4 + x1^3, x2*x1^2 - x1^3, x2^2 - x1^2, y1*x1^3 + y1*x1^2, y1*x2 + y1*x1^2, y1^2 + x1^2, y2*x1 + y1*x1^2]

Note we give no guarantee that the computation of a symmetric Groebner basis will terminate in an order different from lexicographic.

When multiplying Symmetric Ideals or raising them to some integer power, the permutation action is taken into account, so that the product is indeed the product of ideals in the mathematical sense.

sage: I=X*(x[1])
sage: I*I 
Symmetric Ideal (x1^2, x2*x1) of Infinite polynomial ring in x, y over Rational Field
sage: I^3
Symmetric Ideal (x1^3, x2*x1^2, x2^2*x1, x3*x2*x1) of Infinite polynomial ring in x, y over Rational Field
sage: I*I == X*(x[1]^2)
False
__init__(ring, gens, coerce=True)

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]^2+y[2]^2,x[1]*x[2]*y[3]+x[1]*y[4]) # indirect doctest
sage: I
Symmetric Ideal (y2^2 + x1^2, y4*x1 + y3*x2*x1) of Infinite polynomial ring in x, y over Rational Field
sage: from sage.rings.polynomial.symmetric_ideal import SymmetricIdeal
sage: J=SymmetricIdeal(X,[x[1]^2+y[2]^2,x[1]*x[2]*y[3]+x[1]*y[4]])
sage: I==J
True
__mul__(other)

Product of two symmetric ideals.

Since the generators of a symmetric ideal are subject to a permutation action, they in fact stand for a set of polynomials. Hence, when multiplying two symmetric ideals, it does not suffice to simply multiply the respective generators.

EXAMPLE:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1])
sage: I*I         # indirect doctest
Symmetric Ideal (x1^2, x2*x1) of Infinite polynomial ring in x over Rational Field
__pow__(n)

Raise self to some power.

Since the generators of a symmetric ideal are subject to a permutation action, they in fact stand for a set of polynomials. Hence, when raising a symmetric ideals to some power, it does not suffice to simply raise the generators to the respective power.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1])
sage: I^2     # indirect doctest
Symmetric Ideal (x1^2, x2*x1) of Infinite polynomial ring in x over Rational Field
__repr__()

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]^2+y[2]^2,x[1]*x[2]*y[3]+x[1]*y[4])
sage: str(I) # indirect doctest
'Symmetric Ideal (y2^2 + x1^2, y4*x1 + y3*x2*x1) of Infinite polynomial ring in x, y over Rational Field'
_latex_()

EXAMPLES:

sage: from sage.misc.latex import latex
sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]*y[2])
sage: latex(I)
\left(y_{2} x_{1}\right)\Bold{Q}[x_{\ast}, y_{\ast}][\mathfrak{S}_{\infty}]
groebner_basis()

Return a symmetric Groebner basis (type Sequence) of self.

INPUT:

  • tailreduce (optional, default False) - if True, use tail reduction in intermediate computations
  • reduced (optional, default True) - return the reduced normalised Groebner basis
  • algorithm (optional) - determine the algorithm (see below for available algorithms)
  • report (optional) - print information on the progress of computation.
  • use_full_group (optional, default False) - if True then proceed as originally suggested by [AB2008]. Our default method should be faster, see symmetrisation() for more details.

The computation of symmetric Groebner bases also involves the computation of classical Groebner bases, i.e., of Groebner bases for ideals in polynomial rings with finitely many variables. For these computations, Sage provides the following ALGORITHMS:

‘’
autoselect (default)
‘singular:groebner’
Singular’s groebner command
‘singular:std’
Singular’s std command
‘singular:stdhilb’
Singular’s stdhib command
‘singular:stdfglm’
Singular’s stdfglm command
‘singular:slimgb’
Singular’s slimgb command
‘libsingular:std’
libSingular’s std command
‘libsingular:slimgb’
libSingular’s slimgb command
‘toy:buchberger’
Sage’s toy/educational buchberger without strategy
‘toy:buchberger2’
Sage’s toy/educational buchberger with strategy
‘toy:d_basis’
Sage’s toy/educational d_basis algorithm
‘macaulay2:gb’
Macaulay2’s gb command (if available)
‘magma:GroebnerBasis’
Magma’s Groebnerbasis command (if available)

If only a system is given - e.g. ‘magma’ - the default algorithm is chosen for that system.

Note

The Singular and libSingular versions of the respective algorithms are identical, but the former calls an external Singular process while the later calls a C function, i.e. the calling overhead is smaller.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I1 = X*(x[1]+x[2],x[1]*x[2])
sage: I1.groebner_basis()
[x1]
sage: I2 = X*(y[1]^2*y[3]+y[1]*x[3])
sage: I2.groebner_basis()
[y1*x2^2 - y1*x2*x1, y2*x2*x1 - y2*x1^2, y2*y1*x2 - y2*y1*x1, y2*y1^2 + y1*x2, y2^2*y1 + y2*x1]

When using the algorithm originally suggested by Aschenbrenner and Hillar, the result is the same, but the computation takes much longer:

sage: I2.groebner_basis(use_full_group=True)
[y1*x2^2 - y1*x2*x1, y2*x2*x1 - y2*x1^2, y2*y1*x2 - y2*y1*x1, y2*y1^2 + y1*x2, y2^2*y1 + y2*x1]

Last, we demonstrate how the report on the progress of computations looks like:

sage: I1.groebner_basis(report=True, reduced=True)
Symmetric interreduction
[1/2]  >
[2/2] : >
[1/2]  >
[2/2]  >
Symmetrise 2 polynomials at level 2
Apply permutations
>
>
Symmetric interreduction
[1/3]  >
[2/3]  >
[3/3] : >
-> 0
[1/2]  >
[2/2]  >
Symmetrisation done
Classical Groebner basis
-> 2 generators
Symmetric interreduction
[1/2]  >
[2/2]  >
Symmetrise 2 polynomials at level 3
Apply permutations
>
>
:>
::>
:>
::>
Symmetric interreduction
[1/4]  >
[2/4] : >
-> 0
[3/4] :: >
-> 0
[4/4] : >
-> 0
[1/1]  >
Apply permutations
:>
:>
:>
Symmetric interreduction
[1/1]  >
Classical Groebner basis
-> 1 generators
Symmetric interreduction
[1/1]  >
Symmetrise 1 polynomials at level 4
Apply permutations
>
:>
:>
>
:>
:>
Symmetric interreduction
[1/2]  >
[2/2] : >
-> 0
[1/1]  >
Symmetric interreduction
[1/1]  >
[x1]
interreduced_basis()

A fully symmetrically reduced generating set (type Sequence) of self.

This does essentially the same as interreduction() with the option ‘tailreduce’, but it returns a Sequence rather than a SymmetricIdeal.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]+x[2],x[1]*x[2])
sage: I.interreduced_basis()
[-x1^2, x2 + x1]
interreduction(tailreduce=True, sorted=False, report=None, RStrat=None)

Return symmetrically interreduced form of self

INPUT:

  • tailreduce (optional) - If True, the interreduction is also performed on the non-leading monomials.
  • sorted (optional) - If True, it is assumed that the generators of self are already increasingly sorted.
  • report (optional) - If not None, some information on the progress of computation is printed
  • RStrat (optional) - A SymmetricReductionStrategy to which the polynomials resulting from the interreduction will be added. If RStrat already contains some polynomials, they will be used in the interreduction. The effect is to compute in a quotient ring.
RETURN:
A Symmetric Ideal J (sorted list of generators) coinciding with self as an ideal, so that any generator is symmetrically reduced w.r.t. the other generators. Note that the leading coefficients of the result are not necessarily 1.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I=X*(x[1]+x[2],x[1]*x[2])
sage: I.interreduction()
Symmetric Ideal (-x1^2, x2 + x1) of Infinite polynomial ring in x over Rational Field

Here, we show the report option:

sage: I.interreduction(report=True)
Symmetric interreduction
[1/2]  >
[2/2] : >
[1/2]  >
[2/2] T[1] >
>
Symmetric Ideal (-x1^2, x2 + x1) of Infinite polynomial ring in x over Rational Field

[m/n] indicates that polynomial number m is considered and the total number of polynomials under consideration is n. ‘-> 0’ is printed if a zero reduction occurred. The rest of the report is as described in sage.rings.polynomial.symmetric_reduction.SymmetricReductionStrategy.reduce().

Last, we demonstrate the use of the optional parameter RStrat:

sage: from sage.rings.polynomial.symmetric_reduction import SymmetricReductionStrategy
sage: R = SymmetricReductionStrategy(X)
sage: R
Symmetric Reduction Strategy in Infinite polynomial ring in x over Rational Field
sage: I.interreduction(RStrat=R)
Symmetric Ideal (-x1^2, x2 + x1) of Infinite polynomial ring in x over Rational Field
sage: R
Symmetric Reduction Strategy in Infinite polynomial ring in x over Rational Field, modulo
    x1^2,
    x2 + x1
sage: R = SymmetricReductionStrategy(X,[x[1]^2])
sage: I.interreduction(RStrat=R)
Symmetric Ideal (x2 + x1) of Infinite polynomial ring in x over Rational Field
normalisation()

Return an ideal that coincides with self, so that all generators have leading coefficient 1.

Possibly occurring zeroes are removed from the generator list.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I = X*(1/2*x[1]+2/3*x[2], 0, 4/5*x[1]*x[2])
sage: I.normalisation()
Symmetric Ideal (x2 + 3/4*x1, x2*x1) of Infinite polynomial ring in x over Rational Field
reduce(I, tailreduce=False)

Symmetric reduction of self by another Symmetric Ideal or list of Infinite Polynomials.

INPUT:

  • I – a Symmetric Ideal or a list of Infinite Polynomials
  • tailreduce (optional) – if it is True, the non-leading terms will be reduced as well.

Reducing an element p of an Infinite Polynomial Ring X by some other element q means the following:

  1. Let M and N be the leading terms of p and q.
  2. Test whether there is a permutation P that does not does not diminish the variable indices occurring in N and preserves their order, so that there is some term T\in X with T N^P = M. If there is no such permutation, return p
  3. Replace p by p-T q^P and continue with step 1.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I = X*(y[1]^2*y[3]+y[1]*x[3])
sage: I.reduce([y[2]^2*y[1]])
Symmetric Ideal (y3*y1^2 + y1*x3) of Infinite polynomial ring in x, y over Rational Field

The preceding is correct, since any permutation that turns y[2]^2*y[1] into a factor of y[1]^2*y[3] interchanges the variable indices 1 and 2 – which is not allowed. However, reduction by y[1]^2*y[2] works, since one can change variable index 1 into 2 and 2 into 3:

sage: I.reduce([y[1]^2*y[2]])
Symmetric Ideal (y1*x3) of Infinite polynomial ring in x, y over Rational Field

The next example shows that tail reduction is not done, unless it is explicitly advised. The input can also be a symmetric ideal:

sage: J = (x[2])*X
sage: I.reduce(J)
Symmetric Ideal (y3*y1^2 + y1*x3) of Infinite polynomial ring in x, y over Rational Field
sage: I.reduce(J, tailreduce=True)
Symmetric Ideal (y3*y1^2) of Infinite polynomial ring in x, y over Rational Field
squeezed()

Reduce the variable indices occurring in self

OUTPUT:
A Symmetric Ideal whose generators are the result of applying squeezed() to the generators of self.
NOTE:
The output describes the same Symmetric Ideal as self.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ,implementation='sparse')
sage: I = X*(x[1000]*y[100],x[50]*y[1000])
sage: I.squeezed()
Symmetric Ideal (y1*x2, y2*x1) of Infinite polynomial ring in x, y over Rational Field
symmetric_basis()

A symmetrised generating set (type Sequence) of self.

This does essentially the same as symmetrisation() with the option ‘tailreduce’, and it returns a Sequence rather than a SymmetricIdeal.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I = X*(x[1]+x[2], x[1]*x[2])
sage: I.symmetric_basis()
[x1^2, x2 + x1]
symmetrisation(N=None, tailreduce=False, report=None, use_full_group=False)

Apply permutations to the generators of self and interreduce

INPUT:

  • N (optional) – apply permutations in Sym(N). Default is the maximal variable index occurring in the generators of self.interreduction().squeezed().
  • tailreduce (optional) – If True, perform tail reductions
  • report (optional) – If not None, report on the progress of computations
  • use_full_group (optional) – If True, apply all elements of Sym(N) to the generators of self (this is what [AB2008] originally suggests). The default is to apply all elementary transpositions to the generators of self.squeezed(), interreduce, and repeat until the result stabilises, which is often much faster than applying all of Sym(N), and we are convinced that both methods yield the same result.
OUTPUT:
A symmetrically interreduced symmetric ideal with respect to which any Sym(N)-translate of a generator of self is symmetrically reducible, where by default N is the maximal variable index that occurs in the generators of self.interreduction().squeezed().
NOTE:
If I is a symmetric ideal whose generators are monomials, then I.symmetrisation() is its reduced Groebner basis. It should be noted that without symmetrisation, monomial generators, in general, do not form a Groebner basis.

EXAMPLES:

sage: X.<x> = InfinitePolynomialRing(QQ)
sage: I = X*(x[1]+x[2], x[1]*x[2])
sage: I.symmetrisation()
Symmetric Ideal (-x1^2, x2 + x1) of Infinite polynomial ring in x over Rational Field
sage: I.symmetrisation(N=3)
Symmetric Ideal (-2*x1) of Infinite polynomial ring in x over Rational Field
sage: I.symmetrisation(N=3, use_full_group=True)
Symmetric Ideal (-2*x1) of Infinite polynomial ring in x over Rational Field

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