Laws of inertia in higher degree binary forms

TL;DR

This paper investigates the representation of real even-degree forms as sums of powers of linear forms, examining the inertia law's validity across different degrees and revealing its limitations.

Contribution

It extends Sylvester's Law of Inertia to binary quartics and demonstrates its failure for binary sextics, highlighting degree-dependent behavior.

Findings

Inertia law holds for binary quartics.

Inertia law fails for binary sextics.

Degree influences the validity of inertia laws.

Abstract

We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of Inertia holds for binary quartics, but fails for binary sextics.