Simplex Subdivisions and Nonnegativity Decision of Forms

TL;DR

This paper introduces a novel approach using simplex subdivisions to decide the nonnegativity of forms, establishing convergence criteria and developing an algorithm for indefinite forms that extends beyond traditional methods.

Contribution

It presents a new perspective on variable substitutions via simplex subdivisions, introduces convergence concepts, and provides an effective algorithm for nonnegativity decision of forms.

Findings

Established a convergence criterion for subdivision sequences.

Proved the relationship between subdivisions and variable substitutions.

Developed an algorithm for deciding indefinite forms with a counter-example.

Abstract

This paper mainly studies nonnegativity decision of forms based on variable substitutions. Unlike existing research, the paper regards simplex subdivisions as new perspectives to study variable substitutions, gives some subdivisions of the simplex T_n, introduces the concept of convergence of the subdivision sequence, and presents a sufficient and necessary condition for the convergent self-similar subdivision sequence. Then the relationships between subdivisions and their corresponding substitutions are established. Moreover, it is proven that if the form F is indefinite on T_n and the sequence of the successive L-substitution sets is convergent, then the sequence of sets {SLS^(m)(F)} is negatively terminating, and an algorithm for deciding indefinite forms with a counter-example is obtained. Thus, various effective substitutions for deciding positive semi-definite forms and indefinite forms are gained, which are beyond the weighted difference substitutions characterized by "difference".