Real Solution Isolation with Multiplicity of Zero-Dimensional Triangular Systems

TL;DR

This paper introduces a new method for isolating real solutions of zero-dimensional triangular polynomial systems that also computes their multiplicities, using interval arithmetic and square-free factorization.

Contribution

It defines a natural concept of multiplicity for solutions and provides an effective algorithm that computes solutions along with their multiplicities.

Findings

Algorithm successfully isolates solutions with multiplicities

Computational results demonstrate effectiveness on literature examples

Method aligns with classical local multiplicity definitions

Abstract

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local (intersection) multiplicity. Then we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coefficients. The computational results on some examples from the literature are presented.