The weighted difference substitutions and Nonnegativity Decision of Forms
For mathematicians working on polynomial nonnegativity, this offers a simpler convergence proof and decision algorithm, but the contribution is incremental over prior work.
The paper provides geometric interpretations of weighted difference substitutions and proves convergence of successive substitution sets, leading to a simpler method for deciding nonnegativity of positive definite forms. An algorithm for indefinite forms with counter-examples is also presented.
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.