On Wednesday, August 5, 2020 at 6:28:42 AM UTC-5, Santanu wrote:
>
> Dear all,
> Consider ideal I=<x0*x1+x2> over the binary field GF(2).
> Then (x2).reduce(I) gives x2. I want it to be x0*x1.
> In fact , I want this kind of reduction always should give quadratic
> polynomial
> (I know that this is possible for my problems).
>
Dear Santanu
Aside from Nils' answer, you may want to read up on Sage's term orderings:
rings.polynomial.term_order?
It's possible that you want a local term ordering. Unlike global term
orderings, in a local ordering 1>t for any monomial t.
TO = TermOrder("negdegrevlex",3)
R = PolynomialRing(GF(2),'x',3,order=TO)
R.inject_variables()
>>>> Defining x0, x1, x2
I = R.ideal([x0*x1+x2])
x2.reduce(I)
>>>> x0*x1
Another possibility is a weighted (global) term ordering. For example, if I
set the weight vector to (1,2,4), then x2>x0*x1, but I can still have
x0^4>x2.
TO = TermOrder('wdegrevlex',(1,2,4))
R = PolynomialRing(GF(2),'x',3,order=TO)
R.inject_variables()
>>>> Defining x0, x1, x2
I = R.ideal([x0*x1+x2])
x2.reduce(I)
>>>> x0*x1
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