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 $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all $779777$ 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