Hi,
No, my polynomials are not homogeneous and are of different degree. Suppose:
R.<x1,x2,x3>=ZZ[]
f1=1+x1+x2+x1*x2
f2=1+x1+x3+x1*x3
I want to find linear dependence of the set polynomials {f1,f2,f1*f2,
x1*f2,x2*f2}
With regards,
Santanu
On 25 April 2010 15:33, Simon King <simon.k...@nuigalway.ie> wrote:
> Hi!
>
> On 25 Apr., 11:41, Santanu Sarkar <sarkar.santanu....@gmail.com>
> wrote:
> > Suppose f1, f2,....,f10 are polynomials over 20 variables over integers.
> > How one can check weather they are linearly independent or not in Sage?
>
> When you talk about linear indepence of polynomials, you probably
> assume that they are all homogeneous of the same degree, do you?
>
> If this is the case, you could use "interreduction" for obtaining a
> basis of the vector space that is spanned by your polynomials - and
> comparing the size of that basis with the number of the given
> polynomials, you can conclude whether the input is linearly
> independent or not.
>
> Here is a small example, that should easily scale to 10 polynomials
> over 20 variables:
>
> We create 3 homogeneous polynomials of degree 2 on 3 variables:
> sage: R.<x,y,z> = ZZ[]
> sage: p = x^2+y^2
> sage: q = y^2-x*y+z^2
> sage: r = x^2+x*y-z^2
>
> We consider the ideal they generate:
> sage: I = [p,q,r]*R
>
> Since the polynomials are homogeneous of the same degree, the
> following is a basis for the vector space spanned by p,q,r:
> sage: I.interreduced_basis()
> [x^2 + y^2, -x*y + y^2 + z^2]
>
> Since len(I.interreduced_basis() is smaller than I.ngens(), the
> generators of I are (linearly) dependent.
>
> Best regards,
> Simon
>
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