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A deterministic polynomial time algorithm for non-commutative rational identity testing

Year: 2015
Type: preprint
Abstract: In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in {\emph non-commuting} variables over $\mathbb{Q}$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time [IQS15] (whether or not randomization is allowed). The algorithm efficiently solves the word for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential-time algorithms prior to this work. The main (and simple) technical contribution of this paper is an analysis of an existing algorithm due to Gurvits which solved the same problem in some special cases. We also extend the algorithm to compute the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation on the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular we explain how some of these sources played an important role in the development of Gurvits' algorithm and in our analysis of it here.
Language: English
Open Access status: green
OpenAlex ID: W2951834959
Cites: 48
Cited by: 16
Related to: 10