33 research outputs found
Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley
We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman]
it was shown that the method of shifted partial derivatives cannot be used to
separate the padded permanent from the determinant. Mulmuley asked if this
"no-go" result could be extended to a model without padding. We prove this is
indeed the case using the iterated matrix multiplication polynomial. We also
provide several examples of polynomials with maximal space of partial
derivatives, including the complete symmetric polynomials. We apply Koszul
flattenings to these polynomials to have the first explicit sequence of
polynomials with symmetric border rank lower bounds higher than the bounds
attainable via partial derivatives.Comment: 18 pages - final version to appear in Theory of Computin
Geometry of Tensors: Open problems and research directions
This is a collection of open problems and research ideas following the
presentations and the discussions of the AGATES Kickoff Workshop held at the
Institute of Mathematics of the Polish Academy of Sciences (IMPAN) and at the
Department of Mathematics of University of Warsaw (MIM UW), September 19-26,
2022.Comment: Comments are welcome. Final version also available at
https://agates.mimuw.edu.pl/index.php/research-reports-and-note
The geometry of discotopes
We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of discotopes, highlighting interesting birational properties which may be investigated using tools from algebraic geometry. When a discotope is the Minkowski sum of two-dimensional discs, the Zariski closure of its set of extreme points is an irreducible hypersurface. In this case, we provide an upper bound for the degree of the hypersurface, drawing connections to the theory of classical determinantal varieties
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
Matrix product states and the quantum max-flow/min-cut conjectures
In this note we discuss the geometry of matrix product states with periodic
boundary conditions and provide three infinite sequences of examples where the
quantum max-flow is strictly less than the quantum min-cut. In the first we fix
the underlying graph to be a 4-cycle and verify a prediction of Hastings that
inequality occurs for infinitely many bond dimensions. In the second we
generalize this result to a 2d-cycle. In the third we show that the 2d-cycle
with periodic boundary conditions gives inequality for all d when all bond
dimensions equal two, namely a gap of at least 2^{d-2} between the quantum
max-flow and the quantum min-cut.Comment: 12 pages, 3 figures - Final version accepted for publication on J.
Math. Phy
Geometry and Representation Theory in the Study of Matrix Rigidity
The notion of matrix rigidity was introduced by L. Valiant in 1977. He proved a theorem that relates the rigidity of a matrix to the complexity of the linear map that it defines, and proposed to use this theorem to prove lower bounds on the complexity of the Discrete Fourier Transform. In this thesis, I study this problem from a geometric point of view. We reduce to the study of an algebraic variety in the space of square matrices that is the union of linear cones over the classical determinantal variety of matrices of rank not higher than a fixed threshold. We discuss approaches to this problem using classical and modern algebraic geometry and representation theory. We determine a formula for the degrees of these cones and we study a method to find defining equations, also exploiting the classical representation theory of the symmetric group
Partially Symmetric Variants of Comon's Problem Via Simultaneous Rank
A symmetric tensor may be regarded as a partially symmetric tensor in several
different ways. These produce different notions of rank for the symmetric
tensor which are related by chains of inequalities. By exploiting algebraic
tools such as apolarity theory, we show how the study of the simultaneous
symmetric rank of partial derivatives of the homogeneous polynomial associated
to the symmetric tensor can be used to prove equalities among different
partially symmetric ranks. This approach aims to understand to what extent the
symmetries of a tensor affect its rank. We apply this to the special cases of
binary forms, ternary and quaternary cubics, monomials, and elementary
symmetric polynomials.Comment: 28 p
Quatroids and Rational Plane Cubics
It is a classical result that there are (irreducible) rational cubic
curves through generic points in , but little is
known about the non-generic cases. The space of -point configurations is
partitioned into strata depending on combinatorial objects we call quatroids, a
higher-order version of representable matroids. We compute all
quatroids on eight distinct points in the plane, which produces a full
description of the stratification. For each stratum, we generate several
invariants, including the number of rational cubics through a generic
configuration. As a byproduct of our investigation, we obtain a collection of
results regarding the base loci of pencils of cubics and positive certificates
for non-rationality.Comment: 34 pages, 11 figures, 5 tables. Comments are welcome