Related papers: On points with algebraically conjugate coordinates…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it…

Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of…

Let $\mathcal{C}$ be a decomposable plane curve over an algebraically closed field $k$ of characteristic 0. That is, $\mathcal{C}$ is defined in $k^2$ by an equation of the form $g(x) = f(y)$, where $g$ and $f$ are polynomials of degree at…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

For each $1\leq i \le n$, let $k_i\geq 1$ and let $\Delta_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ points in $\mathbb{R}^{k_i+1}$. If $A\subseteq [0,1]^{k_1+1}\times \cdots \times [0,1]^{k_n+1}$ is a Lebesgue…

Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. I prove that there exists $\eta > 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = \alpha$ and $\dim_{\mathrm{H}} B =…

We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first…

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\lambda)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\lambda)$ is real and…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When…

Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in…

We show that for a certain class of convex functions $f$, including the exponential functions $x\mapsto e^{\lambda x}$ with $\lambda>0$ a real number, and all the powers $x\mapsto x^\beta$, $x\geq 0$ and $\beta\geq 2$ a real number, with a…

The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of symmetric polynomials which are indexed by a partition $\lambda$ and depend rationally on a parameter $\alpha$. They reduced to the Schur polynomials when $\alpha=1$, and…

A mapping of $k$-bit strings into $n$-bit strings is called an $(\alpha,\beta)$-map if $k$-bit strings which are more than $\alpha k$ apart are mapped to $n$-bit strings that are more than $\beta n$ apart. This is a relaxation of the…

Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such…

We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^\alpha(1-x)^{\beta} P - Q$, has at most $\deg P +\deg Q + 2$ roots in the interval $]0,~1[$. As a consequence, we…

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,…

In 2016, Ellenberg and Gijswijt employed a method of Croot, Lev, and Pach to show that a maximal cap in $AG(n, q)$ has size $O(q^{cn})$ for some $c < 1$. In this paper, we show more generally that if $S$ is a subset of $AG(n, q)$ containing…