Related papers: Polynomials assuming only local prime powers

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

This is a short note that explains a problem on polynomial maps over finite fields for non-experts. The problem is: Do there exist odd polynomial automorphisms over the finite fields with 4,8,16,32,64,... elements? The explanation is very,…

Athanasiadis raised the question whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. In this paper, we confirm this conjecture and prove the real-rootedness of local $h$-polynomials for all the other…

Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v$ that satisfy given pairwise coprimality conditions. We show how this generalises from monic…

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying…

We study bipartite maps on the plane with one infinite face and one face of perimeter 2. At first we consider the problem of their enumeration an then study the connection between the combinatorial structure of a map and the degree of its…

We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic $p>0$, a few points remain…

We describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields.

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$…

This note is an introduction to the properties of stable polynomials in several variables with real or complex coefficients. These polynomials are defined in terms of where the polynomial is non-vanishing. We do not cover well-known topics…

We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with…

Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high…

In all forms of the local Langlands program the abelian category of smooth representations of p-adic groups G in vector spaces over a field k plays a central role. Of particular interest are its finiteness properties. If the field k has…

For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

Let $a$ be an integer and $p$ a prime so that $f(x)=x^p-a$ is irreducible. Write $f^n(x)$ to indicate the $n$-fold composition of $f(x)$ with itself. We study the monogenicity of number fields defined by roots of $f^n(x)$ and give necessary…

Let k be an algebraically closed field of odd characteristic. We describe derivations of a large class of quantizations of affine normal Poisson varieties over k.