Related papers: The descent set polynomial revisited

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof…

This article, complement to the article [Que], deals with some generalizations of Futw\"angler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $\zeta$ a $p$th primitive root of unity, $K:=\Q(\zeta)$…

Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker…

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. Then…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

For each integer $d\ge 4$, we study the sequence of positive integers which are represented by one at least of the cyclotomic binary forms $\Phi_n(X,Y)$, with $n$ a positive integer satisfying $\varphi(n)\ge d$. The case $d=2$ was studied…

This part is made of three sections. In the first section we study the family of polynomials whose roots are 4cos2 k{\pi}, (n \geqslant 3,1 \leqslant k < \frac{n}{2}). We obtain n2 in this manner a family of orthogonal polynomials. This…

A monic polynomial in F_q[t] of degree n over a finite field F_q of odd characteristic can be written as the sum of two irreducible monic elements in F_q[t] of degrees n and n-1 if q is larger than a bound depending only on n. The main tool…

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight.…

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes,…

Let Y_n(t) denote the set of all polynomials over the ring Z which are reducible over the field Q and of degree n>1 and of height not greater than t. We show that the true order of magnitude of |Y_n(t)| equals t^2 log t in the special case…

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue…

In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for…

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over the field of complex numbers, that can be written as a sum of two polynomials which depend on mutually…

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$…

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares…

We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial…

Let $\mu$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic orthogonal polynomials. In this paper we…

The discriminants of certain polynomials related to Chebyshev polynomials factor into the product of two polynomials, one of which has coefficients that are much larger than the other's. Remarkably, these polynomials of dissimilar size have…

We investigate fixed points and cycle types of permutation polynomials and complete permutation polynomials arising from reversed Dickson polynomials of the first kind and second kind over $\mathbb{F}_p$. We also study the permutation…