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 language | English |
---|---|
Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |
Publisher | IEEE Computer Society |
Pages | 881-888 |
Number of pages | 8 |
ISBN (Electronic) | 9781728196213 |
DOIs | |
State | Published - Nov 2020 |
Externally published | Yes |
Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
---|---|
Volume | 2020-November |
ISSN (Print) | 0272-5428 |
Conference
Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
---|---|
Country/Territory | United States |
City | Virtual, Durham |
Period | 16/11/20 → 19/11/20 |
Bibliographical note
Publisher Copyright:© 2020 IEEE.
Keywords
- null cone membership
- polynomial identity testing
- symmetries of algebraic varieties