Rational Univariate Representations of Bivariate Systems and Applications

Abstract

We address the problem of solving systems of two bivariate polynomials of total degree at most $d$ with integer coefficients of maximum bitsize $\tau$. It is known that a linear separating form, that is a linear combination of the variables that takes different values at distinct solutions of the system, can be computed in $\sOB(d^{8}+d^7\tau)$ bit operations (where $O_B$ refers to bit complexities and $\sO$ to complexities where polylogarithmic factors are omitted) and we focus here on the computation of a Rational Univariate Representation (RUR) given a linear separating form. We present an algorithm for computing a RUR with worst-case bit complexity in $\sOB(d^7+d^6\tau)$ and bound the bitsize of its coefficients by $\sO(d^2+d\tau)$. We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with $\sOB(d^{8}+d^7\tau)$ bit operations. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most $d$ and bitsize at most $\tau$) at one real solution of the system in $\sOB(d^{8}+d^7\tau)$ bit operations and at all the $\Theta(d^2)$ {real} solutions in only $O(d)$ times that for one solution.