The SOS Rank of Biquadratic Forms

TL;DR

This work resolves the sos-rank for the smallest nontrivial class of biquadratic forms beyond the already understood cases by proving that any psd $3\times2$ biquadratic form can be expressed as the sum of four squares of bilinear forms. The authors combine algebraic geometry (Segre–Veronese embeddings, projective normality) with real-analytic tools (Rank Theorem, Federer–Sard) to establish surjectivity of a key multiplication map and to show that every such form lies in the image of a bilinear-squared parametrization. Building on prior results for $2\times2$ forms and Calderón’s bounds, they conjecture a linear $m+1$ bound for general $m\ge4$ and discuss implications for sos-rank theory and computational approaches in polynomial optimization. The results sharpen our understanding of sos representations in structured bilinear settings and may inform semidefinite relaxations in related optimization problems.

Abstract

In 1973, Calderón 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 Calderón'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.

Theorems & Definitions (18)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 3.1
• Theorem 3.2: Rank Theorem
• Lemma 3.3
• proof
• Lemma 3.4
• proof