Zero-nonzero and real-nonreal sign determination

TL;DR

This paper introduces algorithms for zero-nonzero and real-nonreal sign determination problems, analyzing their complexity and proposing improvements, with applications in algebraic geometry and polynomial analysis.

Contribution

It presents new algorithms for zero-nonzero and real-nonreal sign determination problems, including complexity analysis and parametric considerations.

Findings

Algorithms for zero-nonzero determination with complexity analysis

Algorithms for real-nonreal sign determination with complexity analysis

Discussion of parametric context applications

Abstract

We consider first the zero-nonzero determination problem, which consists in determining the list of zero-nonzero conditions realized by a finite list of polynomials on a finite set Z included in C^k with C an algebraic closed field. We describe an algorithm to solve the zero-nonzero determination problem and we perform its bit complexity analysis. This algorithm, which is in many ways an adaptation of the methods used to solve the more classical sign determination problem, presents also new ideas which can be used to improve sign determination. Then, we consider the real-nonreal sign determination problem, which deals with both the sign determination and the zero-nonzero determination problem. We describe an algorithm to solve the real-nonreal sign determination problem, we perform its bit complexity analysis and we discuss this problem in a parametric context.