Related papers: Polynomial products modulo primes and applications

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to…

In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon…

We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic…

We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a…

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…

Let $A$ be a Dedekind domain, $K$ the fraction field, $\p$ a non-zero prime ideal of $A$, and $K_\pp$ the completion of $K$ with respect to the $\p$-adic topology. At the input of a monic irreducible separable polynomial, $f(x)\in A[x]$,…

We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the…

We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in…

We consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems,…

This paper studies properties of the integer sequence $\overline{\overline{G}}_n=\prod_{k=0}^n\binom{n}{k}_{\mathbb{Z},\mathbb{N}}$ which is analogous to $\overline{G}_n=\prod_{k=0}^n\binom{n}{k}$, the product of the elements of the $n$-th…

We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas. We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$…

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…

Let $P(\partial_0,\partial_1,...,\partial_n)$ be a PDO on $\symR^{1+n}$ with constant coefficients. It is proved that (i) the real parts of the $\lambda$-roots of the polynomial $P(\lambda,i\xi_1,...,i\xi_n)$ are bounded from above when…

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…

We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. This extends the results of G. Myerson, who considered the cases $m=1$ and $m=2$, and…

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

We obtain a new lower bound on the size of value set f(F_p) of a sparse polynomial f in F_p[X] over a finite field of p elements when p is prime. This bound is uniform with respect of the degree and depends on some natural arithmetic…