On the Complexity of Solving a Bivariate Polynomial System

TL;DR

This paper analyzes the complexity of the BISOLVE algorithm for solving bivariate polynomial systems, showing it computes solutions efficiently with improved bounds and confirming its superior performance over other methods.

Contribution

The paper provides a complexity analysis of BISOLVE, establishing a new upper bound of tilde(n^8 au^2) bit operations, improving previous results significantly.

Findings

BISOLVE computes solutions within tilde(n^8 au^2) bit operations.

The algorithm outperforms previous methods by a factor of at least n^2.

Extensive benchmarks confirm BISOLVE's efficiency and robustness.

Abstract

We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$- and $y$-axes and, then, selects the actual solutions from the so induced candidate set. However, unlike similar algorithms, BISOLVE requires no genericity assumption on the input nor it needs any change of the coordinate system. Furthermore, extensive benchmarks from \cite{bes-bisolve-2011} confirm that the algorithm outperforms state of the art approaches by a large factor. In this work, we show that, for two polynomials $f,g\in\mathbb{Z}[x,y]$ of total degree at most $n$ with integer coefficients bounded by $2^\tau$, BISOLVE computes isolating boxes for all real solutions of the system $f=g=0$ using $\Otilde(n^8\tau^{2})$ bit operations, thereby improving the previous record bound by a factor of at least $n^{2}$.