矩阵李代数的符号行列式恒等式检验与非交换秩
Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras
摘要：
研究奇异矩阵空间的结构是解决符号行列式恒等式检验问题的一个途径。在解决了非交换秩问题（Garg-Gurvits-Oliveira-Wigderson，Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam，Comput.复杂. 2018），自然的下一步是理解非交换秩满的奇异矩阵空间。目前，这种矩阵空间的例子大多是零星的，所以希望以更系统的方式发现它们。本文通过研究在换位子运算下闭的矩阵空间族，即矩阵李代数，向这个方向迈出了一步。一方面，我们证明了复数域上的矩阵李代数产生具有完全非交换秩的奇异矩阵空间。另一方面，我们证明了这类空间的SDIT可以在确定性多项式时间内确定。此外，我们给出了矩阵李代数产生具有奇异性证明的矩阵空间的一个特征，如Lov'asz（B.布拉兹数学会，1989）和拉兹和Wigderson（建设桥梁II，2019）。
英文摘要：
One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).
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