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 τ We suppose known a linear separating form (that is a linear combination of the variables that takes different values at distinct solutions of the system) and focus on the computation of a Rational Univariate Representation (RUR). We present an algorithm for computing a RUR with worst-case bit complexity in OB(d7+d6τ) and bound the bitsize of its coefficients by O(d2+dτ) (where OB refers to bit complexities and O to complexities where polylogarithmic factors are omitted). We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with OB(d8+d7τ) 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 τ) at one real solution of the system in OB(d8+d7τ) bit operations and at all the ϴ(d2) solutions in only O(d) times that for one solution.