Solving bivariate systems using Rational Univariate Representations

Given two coprime polynomials P and Q in Z[x,y] of degree bounded by d and bitsize bounded by τ, we address the problem of solving the system {P,Q}. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.