The weighted difference substitutions and Nonnegativity Decision of Forms

TL;DR

This paper introduces a geometric approach to weighted difference substitutions, proving their convergence and developing a new method to determine the nonnegativity of forms through finite steps, with an algorithm and examples.

Contribution

It presents a novel geometric interpretation of weighted difference substitutions and a simplified, finite-step method for nonnegativity decision of forms, including an algorithm with counter-examples.

Findings

Proved convergence of weighted difference substitution sets.

Established a finite-step method for nonnegativity decision.

Developed an algorithm with counter-examples for indefinite forms.

Abstract

In this paper, we study the weighted difference substitutions from geometrical views. First, we give the geometric meanings of the weighted difference substitutions, and introduce the concept of convergence of the sequence of substitution sets. Then it is proven that the sequence of the successive weighted difference substitution sets is convergent. Based on the convergence of the sequence of the successive weighted difference sets, a new, simpler method to prove that if the form F is positive definite on T_n, then the sequence of sets {SDS^m(F)} is positively terminating is presented, which is different from the one given in [11]. That is, we can decide the nonnegativity of a positive definite form by successively running the weighted difference substitutions finite times. Finally, an algorithm for deciding an indefinite form with a counter-example is obtained, and some examples are listed by using the obtained algorithm.