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

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.