Symbolic determinant identity testing (sdit) is not a null cone problem; and the symmetries of algebraic varieties

Visu Makam, Avi Wigderson

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

The object of study of this paper is the following multi-determinantal algebraic variety, text{SING} {n, m}, which captures the symbolic determinant identity testing (SDIT) problem (a canonical version of the polynomial identity testing (PIT) problem), and plays a central role in algebra, algebraic geometry and computational complexity theory. text{SING} {n, m} is the set of all m-tuples of n times n complex matrices which span only singular matrices. In other words, the determinant of any linear combination of the matrices in such a tuple vanishes. The algorithmic complexity of testing membership in text{SING} {n, m} is a central question in computational complexity. Having almost a trivial probabilistic algorithm, finding an efficient deterministic algorithm is a holy grail of derandomization, and to top it, will imply super-polynomial circuit lower bounds! A sequence of recent works suggests efficient deterministic'geodesic descent' algorithms for memberships in a general class of algebraic varieties, namely the null cones of (reductive) linear group actions. Can such algorithms be used for the problem above? Our main result is negative: Text{SING} {n, m} is not the null cone of any such group action! This stands in stark contrast to a non-commutative analog of this variety (for which such algorithms work), and points to an inherent structural difficulty of text{SING} {n, m}. In other words, we provide a barrier for the attempts of derandomizing SDIT via these algorithms. To prove this result we identify precisely the group of symmetries of text{SING} {n, m}. We find this characterization, and the tools we introduce to prove it, of independent interest. Our characterization significantly generalizes a result of Frobenius for the special case m=1 (namely, computing the symmetries of the determinant). Our proof suggests a general method for determining the symmetries of general algebraic varieties, an algorithmic problem that was hardly studied and we believe is central to algebraic complexity.

Original languageEnglish
Title of host publicationProceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020
PublisherIEEE Computer Society
Pages881-888
Number of pages8
ISBN (Electronic)9781728196213
DOIs
StatePublished - Nov 2020
Externally publishedYes
Event61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States
Duration: 16 Nov 202019 Nov 2020

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2020-November
ISSN (Print)0272-5428

Conference

Conference61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
Country/TerritoryUnited States
CityVirtual, Durham
Period16/11/2019/11/20

Bibliographical note

Publisher Copyright:
© 2020 IEEE.

Keywords

  • null cone membership
  • polynomial identity testing
  • symmetries of algebraic varieties

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