2022/04/22 by Grigoriy Blekherman, Blekherman, Grigoriy, Bogdan Raiţă +5
Computer Science · Mathematics · #Matrix Theory and Algorithms #Point processes and geometric inequalities #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2204.10625
We study quasiconvex quadratic forms on n × 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× 3 matrices can follow only from the extremality of the determinant of its acoustic tensor, using previous work by Buckley--Šivic (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 × 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.