Completeness of the WDS method in Checking Positivity of Integral Forms

TL;DR

This paper establishes the completeness of the WDS method for verifying the positivity of integral forms by providing upper bounds on the number of steps needed for positive definiteness or semidefiniteness proofs.

Contribution

It introduces upper bounds on WDS steps, demonstrating the method's completeness for checking positivity of integral forms.

Findings

Upper bounds for WDS steps in positivity proofs

WDS method is complete for integral forms

Applicable to positive definite and semidefinite forms

Abstract

Examples show that integral forms can be efficiently proved positive semidefinite by the WDS method, but it was unknown that how many steps of substitutions are needed, or furthermore, which integral forms is this method applicable for. In this paper, we give upper bounds of step numbers of WDS required in proving that an integral form is positive definite, positive semidefinite, or not positive semidefinite, thus deducing that the WDS method is complete.