Related papers: A weak regularity lemma for polynomials
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
An open problem in complexity theory is to find the minimal degree of a polynomial representing the $n$-bit OR function modulo composite $m$. This problem is related to understanding the power of circuits with $\text{MOD}_m$ gates where $m$…
Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to…
We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the…
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against…
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…
Previous work of the second author and Wolf showed that given a set $A\subseteq \mathbb{F}_p^n$ of bounded $\textrm{VC}_2$-dimension, there is a high rank quadratic factor $\mathcal{B}$ of bounded complexity such that $A$ is approximately…
The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex…
Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we…
A univariate graph polynomial P(G;X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing.…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
In the present paper we propose generalizations of the regularity and counting lemmas for multidimensional matrices under a finite alphabet. Firstly, we prove a variant of a multidimensional regularity lemma with the help of a translation…
We introduce different notions of polynomial convexity with bounds on degrees of polynomials in $\mathbb C^n$. We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree…
We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings…
The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [4]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the…
In graph theory, the Szemer\'edi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that…
We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…
Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…
We study the higher H\"older regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained…