论文标题
弱和强大的极端生物质量
Weak and Strong Extremal Biquadratics
论文作者
论文摘要
我们在$ n \ times m $矩阵上研究Quasiconvex二次形式，该形式对应于$（n，m）$变量的非负生物质量形式。我们解释了Harutyunyan-Milton（Comm。PureAppl。Math。70（11），2017年）和Harutyunyan-Hovsepyan（Arch。Arch。Anagration。Anat。244，2022）所说的一个猜想。张量，使用Buckley-šivic的先前工作（线性代数应用598，2020）。我们的主要结果是建立一个Harutyunyan-Milton（Comm。PureAppl。Math。70（11），2017年）的猜想，即$ 3 \ times 3 $矩阵非常强大的极端极端质量。我们的主要技术成分是对库纳特（Kunert）的工作的概括 - 雪橇师对极端非负三元六分化的作品（Trans。Amer。Math。Soc。370（6），2018年）。具体而言，我们表明，当且仅当其品种（超过复数）是一个理性曲线并且其所有奇异性是真实的时，它不是正方形的非负三元六元。
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--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 $3 \times 3$ 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.
