Positive definiteness of fourth order three dimensional symmetric tensors

TL;DR

This paper establishes necessary and sufficient conditions for the positive definiteness of 4th order 3-dimensional symmetric tensors with entries of 1 or -1, and derives inequalities for related ternary quartic polynomials.

Contribution

It provides the first complete analytic characterization of positive definiteness for these specific tensors and applies the results to develop inequalities for ternary quartic polynomials.

Findings

Necessary and sufficient conditions for positive definiteness identified.

Derived strict inequalities for ternary quartic homogeneous polynomials.

Characterized positive definiteness in terms of tensor entries.

Abstract

For a 4th order 3-dimensional symmetric tensor with its some entries $1$ or $-1$, we show the analytic sufficient and necessary conditions of its positive definiteness. By applying these conclusions, several strict inequalities is bulit for ternary quartic homogeneous polynomials.