加权差分代换与型的非负性判定

The Weighted Difference Substitutions and Nonnegativity Decision of Forms

本文从几何的视角来研究加权差分代换,引入代换集序列收敛性概念,证明了逐次加权差分代换是收敛的.并据此给出正定型在逐次加权差分代换下可正向终止的一个严格的证明,得到了不定型在逐次加权差分代换下可负向终止并自动输出反例的一个算法.

This paper provides a geometric perspective to study the weighted difference substitutions and introduces the concept of convergence of the sequence of substitution sets.Then it proves the convergence of the sequence of the successive weighted difference substitution sets,from which it is strictly proved that the sequence of the successive weighted difference substitution sets of a positive definite form is positively terminating. Finally,an algorithm for deciding an indefinite form with a counter-example is proposed.