On some higher degree sign-definite multivariate polynomials associated with definite quadratic forms

TL;DR

This paper introduces a higher degree polynomial associated with quadratic forms, providing a new criterion for definiteness based on sign-definiteness of the polynomial, extending classical quadratic form theory.

Contribution

It establishes a novel connection between quadratic forms and higher degree polynomials, offering a new perspective on definiteness criteria.

Findings

Higher degree polynomial associated with quadratic forms

Polynomial's sign-definiteness characterizes form definiteness

Extends classical quadratic form analysis

Abstract

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present paper a certain higher degree polynomial is associated with each quadratic form such that the form is definite if and only if this polynomial is sign-definite.