Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras
| Autor: | Ivanyos, G��bor, Mittal, Tushant, Qiao, Youming |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
polynomial identity testing QA75 Electronic computers. Computer science / számítástechnika számítógéptudomány Computational Complexity (cs.CC) derandomization non-commutative rank Computer Science - Computational Complexity Theory of computation ��� Pseudorandomness and derandomization Lie algebras FOS: Mathematics Representation Theory (math.RT) symbolic determinant Theory of computation ��� Algebraic complexity theory Mathematics - Representation Theory |
| Popis: | 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��sz (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019). LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 87:1-87:21 |
| Databáze: | OpenAIRE |
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