I have to deal with this a lot, and I've gotten used to it but it
annoys me. More support for specialization would be great. I am not
that good with the guts of the polynomial ring code and singular or I
would have dealt with it already. I would be happy to review patches
related to this.
-M. Hampton
On Oct 29, 10:01 am, Simon King <[EMAIL PROTECTED]> wrote:
> Hi!
>
> On Oct 29, 4:34 pm, vpv <[EMAIL PROTECTED]> wrote:
>
> > The dimension of the ideal of the groebner basis of the new system is
> > 4 and not 0? Why?
>
> Since you are still in a ring with 8 (and not 4) variables.
>
> <snip>
>
>
>
> > What i do next i set:
>
> > x[0]=P(0)
> > x[1]=P(1)
>
> > x[4]=P(1)
> > x[5]=P(0)
>
> > Then my new system is:
>
> > sage: e
>
> > [x2*x7 + x2 + x3*x6 + x3*x7 + x3,
> > x3*x7 + x3 + x7,
> > x2*x7 + x6 + 1,
> > x3*x6 + x3*x7 + x3 + x6 + 1,
> > x2*x7 + x3*x6 + x7,
> > x2*x6 + x2*x7 + x3*x7 + x3,
> > x2*x7 + x2 + x3*x7 + x3 + x7,
> > x2*x6 + x3*x6 + x3,
> > x2*x7 + x3 + x6 + x7 + 1,
> > x2*x6 + x2 + x3*x7,
> > x2*x6 + x2 + x3*x6 + x3 + x7]
>
> Apparently x2,x3,... are still generators of your original polynomial
> ring P -- and P has 8 variables, even though the new system of
> polynomials e uses only four of them. Defining a new ring with only
> the "surviving" variables has the expected result:
>
> sage: P.<x2,x3,x6,x7> = PolynomialRing(GF(2), 4, 'x',order='lex')
> sage: e = [x2*x7 + x2 + x3*x6 + x3*x7 + x3,
> ....: x3*x7 + x3 + x7,
> ....: x2*x7 + x6 + 1,
> ....: x3*x6 + x3*x7 + x3 + x6 + 1,
> ....: x2*x7 + x3*x6 + x7,
> ....: x2*x6 + x2*x7 + x3*x7 + x3,
> ....: x2*x7 + x2 + x3*x7 + x3 + x7,
> ....: x2*x6 + x3*x6 + x3,
> ....: x2*x7 + x3 + x6 + x7 + 1,
> ....: x2*x6 + x2 + x3*x7,
> ....: x2*x6 + x2 + x3*x6 + x3 + x7]
> sage: I=ideal(e)
> sage: I.dimension()
> 0
> sage: I2=ideal(I.groebner_basis())
> sage: V=I2.variety()
> sage: V
> [{x2: 0, x7: 0, x6: 1, x3: 0}]
>
> Substitution of variables in a polynomial ideal *and* carrying it to a
> smaller polynomial ring seems to be a common problem.
> The subs-method does the substitution but keeps the ring:
>
> sage: I.subs({x2:0})
> Ideal (x2*x7 + x2 + x3*x6 + x3*x7 + x3, x3*x7 + x3 + x7, x2*x7 + x6 +
> 1, x3*x6 + x3*x7 + x3 + x6 + 1, x2*x7 + x3*x6 + x7, x2*x6 + x2*x7 +
> x3*x7 + x3, x2*x7 + x2 + x3*x7 + x3 + x7, x2*x6 + x3*x6 + x3, x2*x7 +
> x3 + x6 + x7 + 1, x2*x6 + x2 + x3*x7, x2*x6 + x2 + x3*x6 + x3 + x7) of
> Multivariate Polynomial Ring in x2, x3, x6, x7 over Finite Field of
> size 2
>
> Question to everybody:
> Is there a method that substitutes, say, x2=0 and returns an ideal in
> a polynomial ring over x3, x6, x7?
> If not, do you agree that such method (called, say, "specialisation")
> would be useful?
>
> Cheers
> Simon
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