Related papers: La conjecture de Casas Alvero pour les degr\'es $5…

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result…

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor…

We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E^\ast_\kappa}{V(E^\ast_\kappa)}$ derived from the family of Eisenstein series $E^\ast_\kappa$. The precise,…

Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in \mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r \] where $\mathbf{A}_{m,r}$ are real coefficients. In this…

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have…

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or…

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With…

What is the probability for a number field of composite degree $d$ to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the…

Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such…

In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it…

Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $\lambda_0$ there is a zero of the derivative $f'$ in the closed unit…

A real univariate polynomial with all roots real is called hyperbolic. By Descartes' rule of signs for hyperbolic polynomials (HPs) with all coefficients nonvanishing, a HP with $c$ sign changes and $p$ sign preservations in the sequence of…

In this paper, we examine a natural question concerning the divisors of the polynomial x^n-1: "How often does x^n-1 have a divisor of every degree between 1 and n?" In a previous paper, we considered the situation when x^n-1 is factored in…

Recently, Arag\~{a}o, Marciano, and Mendon\c{c}a [\emph{European J. Combin.}, 2025] conjectured that for any graph $G$ on $n$ vertices satisfying $(r-1)(t-1)k < n \le (r-1)(t-1)(k+1)$, the minimum degree condition $\delta(G) \ge n -…

Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects…

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two…

A multivariate polynomial $p(x)=p(x_1,...,x_n)$ is sos-convex if its Hessian $H(x)$ can be factored as $H(x)= M^T(x) M(x)$ with a possibly nonsquare polynomial matrix $M(x)$. It is easy to see that sos-convexity is a sufficient condition…

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.