Weak and Strong Extremal Biquadratics

TL;DR

This paper investigates the properties of quasiconvex quadratic forms on matrices, disproves a prior conjecture about extremality conditions, and establishes a new link between weak and strong extremality in the context of $3 imes 3$ matrices.

Contribution

It disproves a conjecture relating extremality to the determinant of the acoustic tensor and proves that weak extremal quasiconvex quadratics on $3 imes 3$ matrices are actually strong extremal.

Findings

Disproved the conjecture that extremality depends solely on the determinant of the acoustic tensor.

Proved that weak extremal quasiconvex quadratics on $3 imes 3$ matrices are strong extremal.

Characterized extremal nonnegative ternary sextics as those with rational curve varieties and only real singularities.

Abstract

We study quasiconvex quadratic forms on $n \times m$ matrices which correspond to nonnegative biquadratic forms in $(n,m)$ variables. We disprove a conjecture stated by Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) as well as Harutyunyan--Hovsepyan (Arch. Ration. Mech. Anal. 244, 2022) that extremality in the cone of quasiconvex quadratic forms on $3\times 3$ matrices can follow only from the extremality of the determinant of its acoustic tensor, using previous work by Buckley--\v{S}ivic (Linear Algebra Appl. 598, 2020). Our main result is to establish a conjecture of Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) that weak extremal quasiconvex quadratics on $3 \times 3$ matrices are strong extremal. Our main technical ingredient is a generalization of the work of Kunert--Scheiderer on extreme nonnegative ternary sextics (Trans. Amer. Math. Soc. 370(6), 2018). Specifically, we show that a nonnegative ternary sextic, which is not a square, is extremal if and only if its variety (over the complex numbers) is a rational curve and all its singularities are real.