14 research outputs found
Uniform determinantal representations
The problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
Cut Locus of Submanifolds: A Geometric and Topological Viewpoint
Associated to every closed, embedded submanifold of a connected
Riemannian manifold , there is the distance function which measures
the distance of a point in from . We analyze the square of this function
and show that it is Morse-Bott on the complement of the cut locus
of , provided is complete. Moreover, the gradient flow
lines provide a deformation retraction of to . If is
a closed manifold, then we prove that the Thom space of the normal bundle of
is homeomorphic to . We also discuss several interesting
results which are either applications of these or related observations
regarding the theory of cut locus. These results include, but are not limited
to, a computation of the local homology of singular matrices, a classification
of the homotopy type of the cut locus of a homology sphere inside a sphere, a
deformation of the indefinite unitary group to and a
geometric deformation of to which is
different from the Gram-Schmidt retraction.
\bigskip \noindent If a compact Lie group acts on a Riemannian manifold
freely then is a manifold. In addition, if the action is isometric,
then the metric of induces a metric on . We show that if is a
-invariant submanifold of , then the cut locus is
-invariant, and in
. An application of this result to complex projective hypersurfaces has
been provided.Comment: 121 pages, 33 figures, PhD Thesi
Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras
One approach to make progress on the symbolic determinant identity testing
(SDIT) problem is to study the structure of singular matrix spaces. After
settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson,
Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a
natural next step is to understand singular matrix spaces whose non-commutative
rank is full. At present, examples of such matrix spaces are mostly sporadic,
so it is desirable to discover them in a more systematic way.
In this paper, we make a step towards this direction, by studying the family
of matrix spaces that are closed under the commutator operation, that is matrix
Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the
complex number field give rise to singular matrix spaces with full
non-commutative ranks. On the other hand, we show that SDIT of such spaces can
be decided in deterministic polynomial time. Moreover, we give a
characterization for the matrix Lie algebras to yield a matrix space possessing
singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and
Raz and Wigderson (Building Bridges II, 2019).Comment: 23 page