Related papers: Linear Algebraic Method and the Erd\H{o}s-Heilbron…
In this article, we use the Combinatorial Nullstellensatz to give new proofs of the Cauchy-Davenport, the Dias da Silva-Hamidoune and to generalize a previous addition theorem of the author. Precisely, this last result proves that for a set…
In this paper we prove an $n$th root test for series as well as a Cauchy-Hadamard type formula and Abel's' theorem for power series on universally complete Archimedean complex vector lattices. These results are aimed at developing an…
A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in:…
This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove…
The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex…
In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They…
We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level $\ell$" points, in the sense introduced by Arnold-Shimura, given as a polynomial in the numbers of hypersurfaces…
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…
We show that the properties of the lower part of the spectrum of the Helmholtz equation for an heterogeneous system in a finite region in $d$ dimensions, where the solutions to the homogeneous problems are known, can be systematically…
For any $n>1$ and $0<\varepsilon<1/2$, we show the existence of an $n^{O(1)}$-point subset $X$ of $\mathbb{R}^n$ such that any linear map from $(X,\ell_2)$ to $\ell_2^m$ with distortion at most $1+\varepsilon$ must have $m = \Omega(\min\{n,…
The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…
This paper investigates several classical and novel variations of the Erd\H{o}s--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We…
Using standard techniques from combinatorics, model theory, and algebraic geometry, we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns. Also, we point out a counterexample to a…
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
The Douglas-Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility…
A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.
A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in $\{0, 1\}^n$ with diameter $d$ has cardinality at most that of a Hamming ball of radius $d/2$. In this paper, we give an algebraic…
Classification of finite dimensional representations of the q-deformed Heisenberg algebra $H_q(3)$ is made by the help of Clifford algebra of polynomials and generalized Grassmann algebra. Special attention is paid when $q$ is a primitive…
We establish restrictions on the Hilbert function of standard graded Gorenstein algebras with only quadratic relations. Furthermore, we pose some intriguing conjectures and provide evidence for them by proving them in some cases using a…
In this paper, we introduce a decomposition lemma that allows error terms to be expressed using fewer rank-one symmetric matrices than $\frac{n(n+1)}{2}$ within the convex integration scheme of constructing flexible $C^{1,\alpha}$ solutions…