Continued Fraction Expansion of Real Roots of Polynomial Systems

TL;DR

This paper introduces a novel algorithm for isolating real roots of multivariate polynomial systems using continued fraction expansion, homographies, and integer arithmetic, improving efficiency and handling unbounded regions.

Contribution

It generalizes univariate continued fraction algorithms to multivariate systems, extending Vincent's theorem and providing new complexity bounds.

Findings

Algorithm effectively isolates real roots with rational coordinates.

Homographies enable efficient handling of unbounded regions.

Preliminary implementation demonstrates practical viability.

Abstract

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.