The SOS Rank of Biquadratic Forms

TL;DR

This paper investigates the sum-of-squares rank of positive semidefinite biquadratic forms, improving known results for specific cases and proposing a conjecture for the general case.

Contribution

It proves that 3x2 psd biquadratic forms can be expressed as four squares, extending previous results and conjecturing a general formula for m x 2 forms.

Findings

3x2 psd biquadratic forms are sums of four squares

Improves Calderón's result for m=2

Proposes a conjecture for general m x 2 forms

Abstract

In 1973, Calder\'{o}n proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calder\'{o}n's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.