Separating linear forms for bivariate systems

Abstract

We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most~$τ$, our algorithm computes a separating linear form in $\sOB(d^{8}+d^7τ)$ bit operations in the worst case, where the previously known best bit complexity for this problem was $\sOB(d^{10}+d^9τ)$ (where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity).