Separating linear forms for bivariate systems

• Authors: Yacine Bouzidi;Sylvain Lazard;Marc Pouget;Fabrice Rouillier
• Venue: Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
• Year: 2013

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 of 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 separatin linear form in ÕB(d8+d7τ+d5τ2) bit operations in the worst case, where the previously known best bit complexity for this problem was ÕB(d10+d9τ) (whereÕ refers to the complexity where polylogarithmic factors are omitted and ÕB refers to the bit complexity)