#5566: [with patch, needs review] Groebner bases of Symmetric Ideals
---------------------------------+------------------------------------------
Reporter: SimonKing | Owner: SimonKing
Type: enhancement | Status: new
Priority: major | Milestone: sage-3.4.1
Component: commutative algebra | Keywords: Symmetric Ideal
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Comment(by SimonKing):
I think that now my implementation is in a shape that allows for
applications. In fact, i already computed a Symmetric Groebner Basis that
should provide a topological invariant for knots and links.
As i announced above, i made variables of index zero ''inert'' against
permutations / permutation group elements. So, we have
{{{
sage: X.<x,y> = SymmetricPolynomialRing(QQ)
sage: P = SymmetricGroup(5).random_element()
sage: x[0]^P
x0
}}}
Compared with my first implementation, reduction is now a lot faster. This
is since i now use the usual libsingular-reduction as much as possible.
Another trick is that i try to find a compromise between the requirement
of small parents versus the feature of having a common parent for all
underlying finite polynomials: While computing with symmetric ideals, in
some cases a common parent is found for the underlying polynomials of all
generators of the ideal.
Example: if we start with an ideal
{{{
sage: I = X*(x[1]*y[1],x[50]*y[1000])
}}}
then we will have
{{{
sage: [p._p.parent() for p in I.gens()] # start with different rings
[Multivariate Polynomial Ring in x1, y1 over Rational Field,
Multivariate Polynomial Ring in x50, y1000 over Rational Field]
sage: J=I.groebner_basis()
sage: J
Ideal (x1*y1, x50*y1000, x51*y1) of Symmetric polynomial ring in y, x over
Rational Field
sage: [p._p.parent() for p in J.gens()] # end with common rings
[Multivariate Polynomial Ring in x51, x50, x1, y1000, y1 over Rational
Field,
Multivariate Polynomial Ring in x51, x50, x1, y1000, y1 over Rational
Field,
Multivariate Polynomial Ring in x51, x50, x1, y1000, y1 over Rational
Field]
}}}
In that way, we make arithmetic fast (<= common ring) and slim at the same
time.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5566#comment:6>
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