Existence of positive definite noncoercive sums of squares

TL;DR

This paper constructs positive definite sums of squares with the property that any collection of forms summing to it must share a common nontrivial complex root, revealing new structural insights.

Contribution

It introduces a novel class of positive definite sums of squares with a shared root property, expanding understanding of sum of squares representations.

Findings

Constructed explicit examples of such sums of squares.

Proved that all summing forms share a common nontrivial complex root.

Enhanced the theoretical framework of positive definite forms.

Abstract

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.