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:
Ideal in an Infinite Polynomial Ring, invariant under permutation of variable indices
THEORY:
An Infinite Polynomial Ring with finitely many generators over
a field
is a free commutative
-algebra generated by infinitely many ‘variables’
. We refer to the natural number
as the index
of the variable
. See more detailed description at infinite_polynomial_ring
Infinite Polynomial Rings are equipped with a permutation action by permuting positive
variable indices, i.e., for any permutation
.
Note that the variables
of index zero are invariant under that action.
A Symmetric Ideal is an ideal in an infinite polynomial ring that is invariant under
the permutation action. In other words, if
denotes the symmetric
group of
, then a Symmetric Ideal is a right
-submodule
of
.
It is known by work of Aschenbrenner and Hillar [AB2007] that an Infinite Polynomial
Ring with a single generator
is Noetherian, in the sense
that any Symmetric Ideal
is finitely generated modulo addition,
multiplication by elements of
, and permutation of variable indices (hence, it is a
finitely generated right
-module).
Moreover, if is equipped with a lexicographic monomial ordering with
then there is an algorithm of Buchberger type that computes a Groebner basis
for
that allows for computation of a unique normal form, that is zero precisely for the elements
of
– 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
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
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
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
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'
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}]
Return a symmetric Groebner basis (type Sequence) of self.
INPUT:
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:
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]
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]
Return symmetrically interreduced form of self
INPUT:
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
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
Symmetric reduction of self by another Symmetric Ideal or list of Infinite Polynomials.
INPUT:
Reducing an element of an Infinite Polynomial Ring
by some other element
means the following:
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
Reduce the variable indices occurring in 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
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]
Apply permutations to the generators of self and interreduce
INPUT:
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