2026-07-22T11:01:32Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15683 2024-03-06T10:55:56Z ddc:004 open_access

Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras Ivanyos, Gábor Mittal, Tushant Qiao, Youming derandomization polynomial identity testing symbolic determinant non-commutative rank Lie algebras 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 Lovász (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gábor Ivanyos and Tushant Mittal and Youming Qiao 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.87 urn:nbn:de:0030-drops-156837 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.87 eng https://creativecommons.org/licenses/by/4.0/legalcode