Related papers: Factorization Statistics of Restricted Polynomial …

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations we define, for each prime $\ell$, a local factor which measures the relative frequency with which $f\bmod \ell$ occurs as the characteristic polynomial…

We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…

We determine the factorization of X*f(X)-Y*g(Y) over K[X,Y] for all squarefree additive polynomials f,g in K[X] and all fields K of odd characteristic. This answers a question of Kaloyan Slavov, who needed these factorizations in connection…

We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges…

For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this…

In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1,…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field…

In this paper we study scalar multivariate subdivision schemes with general integer expanding dilation matrix. Our main result yields simple algebraic conditions on the symbols of such schemes that characterize their polynomial…

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions…

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k + \ldots + \alpha_1 x + \alpha_0,$$ where $\alpha_0, \ldots , \alpha_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the…

The general stability problem of truncations for a family of functions concentrating mass at the origin is described and a concrete example in the framework of entire optimizers for the fractional Hardy-Sobolev inequality is given. In this…

We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and P(n) is the largest prime factor of $n$.…

We introduce the notion of semi-characteristic polynomial for a semi-linear map of a finite- dimensional vector space over a field of characteristic p. This polynomial has some properties in common with the classical characteristic…

This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N…

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0,…

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number greater or equal than 5. In this aim, we use the symmetric Chudnovsky-type…