Related papers: Towards the classification of exceptional scattere…

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic…

In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when…

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduced this concept and study when…

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal…

The $n$-grid $E_n$ consists of $n$ equally spaced points in $[-1,1]$ including the endpoints $\pm 1$. The extremal polynomial $p_n^*$ is the polynomial that maximizes the uniform norm $\| p \|_{[-1,1]}$ among polynomials $p$ of degree $\leq…

We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$ These are the first known upper bounds in the polynomial Szemer\'{e}di theorem for the…

We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight…

For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient…

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for $q$, a power of an odd prime, and…

This article studies exponential families $\mathcal{E}$ on finite sets such that the information divergence $D(P\|\mathcal{E})$ of an arbitrary probability distribution from $\mathcal{E}$ is bounded by some constant $D>0$. A particular…

Let $k$ be a number field. We study how well can finite sets of $\mathcal O_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly…

Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any…

Building on the work of Arizmendi and Celestino (2021), we derive the $*$-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $\alpha$ and $\beta$, we derive…

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is…

This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…