Improved algorithm for computing separating linear forms for bivariate systems

TL;DR

This paper introduces a new, more efficient algorithm for computing linear separating forms in bivariate polynomial systems, significantly reducing the worst-case bit complexity compared to previous methods.

Contribution

The paper presents a novel algorithm with improved worst-case and expected bit complexity for computing separating linear forms in bivariate systems.

Findings

Reduces worst-case bit complexity by a factor of d

Provides a probabilistic Las-Vegas algorithm with lower expected complexity

Simplifies the computation process compared to previous algorithms

Abstract

We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. 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 this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity $\sOB(d^7+d^6\tau)$ where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity $\sOB(d^5+d^4\tau)$.