Affiliation:
1. School of Mathematics and Statistics , University of Melbourne , Melbourne , Australia
2. School of Mathematics , Institute for Advanced Study , Princeton , USA
Abstract
Abstract
The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory:
SING
n
,
m
{{\rm SING}_{n,m}}
, consisting of all m-tuples of
n
×
n
{n\times n}
complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in
SING
n
,
m
{{\rm SING}_{n,m}}
will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation.
A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative:
SING
n
,
m
{{\rm SING}_{n,m}}
is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of
SING
n
,
m
{{\rm SING}_{n,m}}
.
To prove this result, we identify precisely the group of symmetries of
SING
n
,
m
{{\rm SING}_{n,m}}
. We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case
m
=
1
{m=1}
, and suggests a general method for determining the symmetries of algebraic varieties.
Subject
Applied Mathematics,General Mathematics
Reference53 articles.
1. Z. Allen-Zhu, A. Garg, Y. Li, R. Oliveira and A. Wigderson,
Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing,
STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing,
ACM, New York (2018), 172–181.
2. S. A. Amitsur,
Rational identities and applications to algebra and geometry,
J. Algebra 3 (1966), 304–359.83]
3. P. Bürgisser, C. Franks, A. Garg, R. Oliveira, M. Walter and A. Wigderson,
Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes,
59th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2018,
IEEE Computer Society, Los Alamitos (2018), 883–897.
4. P. Bürgisser, A. Garg, R. Oliveira, M. Walter and A. Wigderson,
Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory,
9th Innovations in Theoretical Computer Science,
LIPIcs. Leibniz Int. Proc. Inform. 94,
Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Wadern (2018), Article No. 24.
5. P. Bürgisser, M. Levent Doğan, V. Makam, M. Walter and A. Wigderson,
Polynomial time algorithms in invariant theory for torus actions,
preprint (2021), https://arxiv.org/abs/2102.07727.
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