Related papers: Correlations between real conjugate algebraic numb…

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $\Phi_n(q)^2$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward…

In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

An unresolved conjecture by Graham Higman states that for all $n\geq 1$ the number of conjugacy classes of the group of $n \times n$ unitriangular matrices with entries in the finite field $\mathbb{F}_q$ is a polynomial in $q$. In this…

We give a new and simple proof of a theorem of Garza estimating the height (or Mahler measure) of an algebraic number with real conjugates.

We study some elementary properties of the quantum enveloping algebra associated to a parabolic subalgebra $\mathfrak{p}$ of a semisimple Lie algebra $\mathfrak{g}$. In particular we prove an explicit formula for the degree of this algebra,…

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$. We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that…

We have recently show that Poincare series of Hyperbolic Lie algebras have the form of a ratio between Poincare series of a chosen finite Lie algebra and a polynomial of finite degree. By the aid of some properly chosen examples, we now…

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map \Phi_p from the set \underline{W} of conjugacy classes in the Weyl group W to the set of unipotent…

Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of…

The profile of a relational structure R is the function phi_R which counts for every integer n the number, possibly infinite, phi_R(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified.…

Let $A\subseteq\{1,...,N\}$ and $P_1,...,P_\ell\in\Z[n]$ with $P_i(0)=0$ and $\deg P_i=k$ for every $1\leq i\leq\ell$. We show, using Fourier analytic techniques, that for every $\VE>0$, there necessarily exists $n\in\N$ such that…

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function…

Let $X$ be a smooth proper curve defined over a field $k$. The representability of the relative Picard functor is obstructed by a class $\alpha\in\mathrm{Br}(\mathrm{Pic}_{X/k})$. We show the associated division algebra on…

In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a…

We prove that the sequence $(1/F_{n+2})_{n\ge 0}$ of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little $q$-Jacobi polynomials with…

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k}…

Given a power $q$ of a prime number $p$ and "nice" polynomials $f_1,...,f_r\in\bbF_q[T,X]$ with $r=1$ if $p=2$, we establish an asymptotic formula for the number of pairs $(a_1,a_2)\in\bbF_q^2$ such that…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set,…