Help with algorithm to find rational roots of a bivariate equation?

*To*: mathgroup at smc.vnet.net*Subject*: [mg105080] Help with algorithm to find rational roots of a bivariate equation?*From*: TPiezas <tpiezas at gmail.com>*Date*: Fri, 20 Nov 2009 06:39:05 -0500 (EST)

Hello all, Does anyone know an efficient algorithm using Mathematica that can find _rational_ roots of a non-homogenous eqn in two variables with deg > 4? For ex, you want to find rational {x,y} such that, F(x,y) = x^n + (P_1)x^(n-1) + (P_2)x^(n-2) + .... = 0 where the P_i are polynomials in y. I _always_ come across this situation in the course of experimental mathematics, and it would be great if Mathematica had a built-in feature that solves _bivariate_ eqns in the rationals. Right now, I have a 22-deg eqn in x with coefficients in y. I know three non- trivial rational pairs {x,y} such that F(x,y) = 0, but I want to know if there are others. If there are, a certain family of identities would have more members. Any help will be appreciated. - Titus

**Follow-Ups**:**Re: Help with algorithm to find rational roots of a bivariate***From:*Daniel Lichtblau <danl@wolfram.com>

**Re: Help with algorithm to find rational roots of a bivariate equation?***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>