An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

TL;DR

This paper introduces an exact, complete elimination algorithm for solving zero-dimensional bivariate polynomial systems that reduces symbolic operations, avoids generic position assumptions, and leverages hardware acceleration for improved efficiency.

Contribution

The paper presents a novel elimination method that minimizes symbolic computations, does not require coordinate transformations, and incorporates hardware acceleration and filtering techniques.

Findings

Outperforms state-of-the-art implementations on challenging benchmarks

Reduces symbolic operations to resultants and square-free factorization

Efficiently isolates real solutions without generic position assumptions

Abstract

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.