Related papers: On vanishing sums of $\,m\,$th roots of unity in f…

For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the…

Consider the $m$-th roots of unity in {\bf C}, where $m>0$ is an integer. We address the following question: For what values of $n$ can one find $n$ such $m$-th roots of unity (with repetitions allowed) adding up to zero? We prove that the…

A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the…

We study the zeros of cusp forms in the Miller basis whose vanishing order at infinity is a fixed number $m$. We show that for sufficiently large weights, the finite zeros of such forms in the fundamental domain, all lie on the circular…

Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive…

Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$…

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…

The classical problem of whether $m$th-powers with or without zero in a finite field $\mathbb{F}_q$ form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this…

We consider the problem of whether or not certain mock theta functions vanish at the roots of unity with an odd order. We prove for any such function $f(q)$ that there exists a constant $C>0$ such that for any odd integer $n>C$ the function…

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.

In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a…

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we use the method developed before to solve one more special case. We make extensive use of standard…

We completely classify Laurent series converging on the unit circle over a non-Archimedean local field (of any characteristic) that map infinitely many roots of unity to roots of unity. For a given Laurent series $f$ over a field of…

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

We study properties of positive functions satisfying (E) --$\Delta$u + u p -- M |$\nabla$u| q = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the…

Given an integer $M\geq 2$, we deploy the generating function techniques to compute the number of $M$-th roots of identity in some of the well-known finite groups of Lie type, more precisely for finite general linear groups, symplectic…