Related papers: Squarefree polynomials and Mobius values in short …

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

An exponent of distribution 1/16 is established for square-free palindromes. The main input is an upper bound for the number of palindromes, in arithmetic progressions to large moduli, divisible by large squares. Our argument combines a…

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$,…

This paper is concerned with squarefree values of polynomials and their density in large boxes centered at the origin.

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…

Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new…

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N<x\le 2N$) of the "symmetry"…

We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of…

This paper studies long range random walks on ${\mathbb{Z}_q}^d$. $X_{t+1} = X_t + Z_t \mod q$, with $(Z_t)$ independent and identically distributed. Multiple entries of $Z_t$ can be non-zero in a transition. An emphasis is on finding the…

Let $F({\bf x})\in\mathbb{Z}[x_1,x_2,\dots,x_n]$ be a quadratic polynomial in $n\geq 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $$ \Sigma_{k,F}(X;…

Asymptotic behavior of orthogonal polynomials on the circle, with respect to a weight having a fractional zero on the torus. Applications to the eigenvalues of certain unitary random matrices. This paper is devoted to the orthogonal…

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion of integrals, and…

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 $\log n$.…

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $\alpha$ of $f$ in the algebraic closure of $F$,…

We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper…

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…

We prove quantitative estimates for averages of the von Mangoldt and M\"obius functions along polynomial progressions $n+P_1(m),\ldots, n+P_k(m)$ for a large class of polynomials $P_i$. The error terms obtained save an arbitrary power of…

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only…

In this paper, we study different generalizations of the notion of squarefreeness for ideals to the more general case of modules. We describe the cones of Hilbert functions for squarefree modules in general and those generated in degree…