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:

• Simon King <simon.king@nuigalway.ie>

EXAMPLES:

Here, we demonstrate that working in quotient rings of Infinite Polynomial Rings works, provided that one uses symmetric Groebner bases.

sage: R.<x> = InfinitePolynomialRing(QQ)
sage: I = R.ideal([x[1]*x[2] + x[3]])

Note that I is not a symmetric Groebner basis:

sage: G = R*I.groebner_basis()
sage: G
Symmetric Ideal (x_1^2 + x_1, x_2 - x_1) of Infinite polynomial ring in x over Rational Field
sage: Q = R.quotient(G)
sage: p = x[3]*x[1]+x[2]^2+3
sage: Q(p)
-2*x_1 + 3

By the second generator of G, variable \(x_n\) is equal to \(x_1\) for any positive integer \(n\). By the first generator of G, \(x_1^3\) is equal to \(x_1\) in Q. Indeed, we have

sage: Q(p)*x[2] == Q(p)*x[1]*x[3]*x[5]
True

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

Bases: Ideal_generic

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.

EXAMPLES:

sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: I = [x[1]*y[2]*y[1] + 2*x[1]*y[2]]*X
sage: I == loads(dumps(I))
True
sage: latex(I)
\left(x_{1} y_{2} y_{1} + 2 x_{1} y_{2}\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() ; J   # about 3 seconds
[x_1*y_2*y_1 + 2*x_1*y_2, x_2*y_2*y_1 + 2*x_2*y_1, x_2*x_1*y_1^2 + 2*x_2*x_1*y_1, x_2*x_1*y_2 - x_2*x_1*y_1]

Note that even though the symmetric ideal can be generated by a single polynomial, its reduced symmetric Groebner basis comprises four elements. Ideal membership in I can now be tested by commuting symmetric reduction modulo J:

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

The Groebner basis is not point-wise invariant under permutation:

sage: P=Permutation([2, 1])
sage: J[2]
x_2*x_1*y_1^2 + 2*x_2*x_1*y_1
sage: J[2]^P
x_2*x_1*y_2^2 + 2*x_2*x_1*y_2
sage: J[2]^P in J
False

However, any element of J has symmetric reduction zero even after applying a permutation. This even holds when the permutations involve higher variable indices than the ones occurring in J:

sage: [[(p^P).reduce(J) for p in J] for P in Permutations(3)]
[[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 cannot detect ideal membership:

sage: [p.reduce(I) for p in J]
[0, x_2*y_2*y_1 + 2*x_2*y_1, x_2*x_1*y_1^2 + 2*x_2*x_1*y_1, x_2*x_1*y_2 - x_2*x_1*y_1]

Note that we give no guarantee that the computation of a symmetric Groebner basis will terminate in any 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 (x_1^2, x_2*x_1) of Infinite polynomial ring in x, y over Rational Field
sage: I^3
Symmetric Ideal (x_1^3, x_2*x_1^2, x_2^2*x_1, x_3*x_2*x_1) of Infinite polynomial ring in x, y over Rational Field
sage: I*I == X*(x[1]^2)
False

groebner_basis(tailreduce=False, reduced=True, algorithm=None, report=None, use_full_group=False)

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

INPUT:

• tailreduce – (bool, default False) If True, use tail reduction in intermediate computations

• reduced – (bool, default True) If True, return the reduced normalised symmetric Groebner basis.

• algorithm – (string, default None) Determine the algorithm (see below for available algorithms).

• report – (object, default None) If not None, print information on the progress of computation.

• use_full_group – (bool, 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()
[x_1]
sage: I2 = X*(y[1]^2*y[3]+y[1]*x[3])
sage: I2.groebner_basis()
[x_1*y_2 + y_2^2*y_1, x_2*y_1 + y_2*y_1^2]

Note that a symmetric Groebner basis of a principal ideal is not necessarily formed by a single polynomial.

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)
[x_1*y_2 + y_2^2*y_1, x_2*y_1 + y_2*y_1^2]

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]  >
[x_1]

The Aschenbrenner-Hillar algorithm is only guaranteed to work if the base ring is a field. So, we raise a TypeError if this is not the case:

sage: R.<x,y> = InfinitePolynomialRing(ZZ)
sage: I = R*[x[1]+x[2],y[1]]
sage: I.groebner_basis()
Traceback (most recent call last):
...
TypeError: The base ring (= Integer Ring) must be a field

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()
[-x_1^2, x_2 + x_1]

interreduction(tailreduce=True, sorted=False, report=None, RStrat=None)

Return symmetrically interreduced form of self

INPUT:

• tailreduce – (bool, default True) If True, the interreduction is also performed on the non-leading monomials.

• sorted – (bool, default False) If True, it is assumed that the generators of self are already increasingly sorted.

• report – (object, default None) If not None, some information on the progress of computation is printed

• RStrat – (SymmetricReductionStrategy, default None) A reduction strategy 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.

OUTPUT:

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 (-x_1^2, x_2 + x_1) 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 (-x_1^2, x_2 + x_1) 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 (-x_1^2, x_2 + x_1) of Infinite polynomial ring in x over Rational Field
sage: R
Symmetric Reduction Strategy in Infinite polynomial ring in x over Rational Field, modulo
    x_1^2,
    x_2 + x_1
sage: R = SymmetricReductionStrategy(X,[x[1]^2])
sage: I.interreduction(RStrat=R)
Symmetric Ideal (x_2 + x_1) of Infinite polynomial ring in x over Rational Field

is_maximal()

Answers whether self is a maximal ideal.

ASSUMPTION:

self is defined by a symmetric Groebner basis.

NOTE:

It is not checked whether self is in fact a symmetric Groebner basis. A wrong answer can result if this assumption does not hold. A NotImplementedError is raised if the base ring is not a field, since symmetric Groebner bases are not implemented in this setting.

EXAMPLES:

sage: R.<x,y> = InfinitePolynomialRing(QQ)
sage: I = R.ideal([x[1]+y[2], x[2]-y[1]])
sage: I = R*I.groebner_basis()
sage: I
Symmetric Ideal (y_1, x_1) of Infinite polynomial ring in x, y over Rational Field
sage: I = R.ideal([x[1]+y[2], x[2]-y[1]])
sage: I.is_maximal()
False

The preceding answer is wrong, since it is not the case that I is given by a symmetric Groebner basis:

sage: I = R*I.groebner_basis()
sage: I
Symmetric Ideal (y_1, x_1) of Infinite polynomial ring in x, y over Rational Field
sage: I.is_maximal()
True

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 (x_2 + 3/4*x_1, x_2*x_1) 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, or symmetric reduction of a given Infinite Polynomial by self.

INPUT:

• I – an Infinite Polynomial, or a Symmetric Ideal or a list of Infinite Polynomials.

• tailreduce – (bool, default False) If True, the non-leading terms will be reduced as well.

OUTPUT:

Symmetric reduction of self with respect to I.

THEORY:

Reduction of 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]^2)
sage: I.reduce([x[1]^2*y[2]])
Symmetric Ideal (x_3^2*y_1 + y_3*y_1^2) of Infinite polynomial ring in x, y over Rational Field

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

sage: I.reduce([x[2]^2*y[1]])
Symmetric Ideal (y_3*y_1^2) 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 = (y[2])*X
sage: I.reduce(J)
Symmetric Ideal (x_3^2*y_1 + y_3*y_1^2) of Infinite polynomial ring in x, y over Rational Field
sage: I.reduce(J, tailreduce=True)
Symmetric Ideal (x_3^2*y_1) 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 (x_2*y_1, x_1*y_2) 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()
[x_1^2, x_2 + x_1]

symmetrisation(N=None, tailreduce=False, report=None, use_full_group=False)

Apply permutations to the generators of self and interreduce

INPUT:

• N – (integer, default None) Apply permutations in \(Sym(N)\). If it is not given then it will be replaced by the maximal variable index occurring in the generators of self.interreduction().squeezed().

• tailreduce – (bool, default False) If True, perform tail reductions.

• report – (object, default None) 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 (-x_1^2, x_2 + x_1) of Infinite polynomial ring in x over Rational Field
sage: I.symmetrisation(N=3)
Symmetric Ideal (-2*x_1) of Infinite polynomial ring in x over Rational Field
sage: I.symmetrisation(N=3, use_full_group=True)
Symmetric Ideal (-2*x_1) of Infinite polynomial ring in x over Rational Field