Related papers: Quasi-random multilinear polynomials

Let $F$ be a family of pseudo-disks in the plane, and $P$ be a finite subset of $F$. Consider the hypergraph $H(P,F)$ whose vertices are the pseudo-disks in $P$ and the edges are all subsets of $P$ of the form $\{D \in P \mid D \cap S \neq…

In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this…

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…

We revisit the Bohnenblust--Hille multilinear and polynomial inequalities and prove some new properties. Our main result is a multilinear version of a recent result on polynomials whose monomials have a uniformly bounded number of…

This work investigates the invariance of the non-necessarily finite uniform dimension and related concepts for subextensions in skew polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}; \mathbf{\alpha} , \mathbf{\delta} ]$} of…

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such…

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $\Gamma_1,\ldots,\Gamma_j$ of low-degree…

This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$,…

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…

The $\epsilon$-approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within $\epsilon$ in the $\ell_\infty$ norm. We prove several lower bounds on this…

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend…

We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous…

The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem,…

We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on…

We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition…

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods…