Related papers: Distribution of complex algebraic numbers on the u…

In this work in progress, we study the asymptotic behaviour of the $p$-quantile of the Beta distribution, i.e. the quantity $q$ defined implicitly by $\int_0^q t^{a - 1} (1 - t)^{b - 1} \text{d} t = p B (a, b)$, as a function of the first…

The spectrum of a real number $\beta>1$ is the set $X^{m}(\beta)$ of $p(\beta)$ where $p$ ranges over all polynomials with coefficients restricted to ${\mathcal A}=\{0,1,\dots,m\}$. For a quadratic Pisot unit $\beta$, we determine the…

Let $q=2^n$ with $n=2m$ . Let $1\leq k\leq n-1$ and $k\neq m$. In this paper we determine the value distribution of following exponential sums \[\sum\limits_{x\in \bF_q}(-1)^{\Tra_1^m (\alpha x^{2^{m}+1})+\Tra_1^n(\beta…

For every positive integer $n$ and every $\delta \in [0,1]$, let $B(n, \delta)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$…

In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We…

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $\beta$ such that $|\beta|>1$, the number $[\![\boldsymbol{u} ]\!]_\beta:=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$…

In this paper we study the distribution of the size of the value set for a random polynomial with degree at most $q-1$ over a finite field $\mathbb{F}_q$. We obtain the exact probability distribution and show that the number of missing…

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P)…

We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-\Delta)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under…

This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are…

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2\pi$-periodic functions, which are represented by convolutions of functions $\varphi (\varphi\bot 1)$ from unit ball of…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

In this note we introduce a new class of refined Eulerian polynomials defined by $$A_n(p,q)=\sum_{\pi\in\mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},$$ where ${\rm odes}(\pi)$ and ${\rm edes}(\pi)$ enumerate the number of descents…

Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a…

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime…

The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer…

In this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2n all of whose roots lie on the unit circle. In particular, we consider a set V_n of vectors in R^n…

We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<…