Bill Allombert on Mon, 02 Feb 2009 17:42:46 +0100

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Re: System of two polynomial equations

On Mon, Feb 02, 2009 at 04:18:48PM +0100, wrote:
>              Hello all,
>      if F and G are two polynomials in two variables x
> and y, the system {F(x,y)=0,G(x,y)=0}, is usually
> equivalent to a system of the form {P(x)=0,y=Q(x)}, where
> P and Q are univariate polynomials.

Could you provide an example and a more definite statement ?
Obviously this imply that there are no pair of solution of the form
{(x0,y0),(x0,y1)} with y0!=y1 because y0=Q(x0)=y1.

>   P can be easily computed with GP: it's essentially equal to
> polresultant(F(x,y),G(x,y),y).
>    What about Q ? I can't think of a GP function that deals with
> this in a simple way.

I suggest you factorize P over K and then factorize G over (K[x]/P).
In case where K=Q, you can use nffactor to factorize Q.
In case where K=C, you can use polroots.