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The aim of this work is to use Napoleon's Theorem in different regular polygons, and decide whether we can prove Napoleon's Theorem is only limited with triangles or it could be done in other regular polygons that can create regular…
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…
In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree.…
Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons -- or even to formalize what a random polygon is -- it is convenient to interpret each polygon as a point…
Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of…
Let f(x) be a polynomial with integer coefficients, let n be a positive integer, and let p be an odd prime. Then the mapping x-->f(x) sends Z/p^n into Z/p^n. We study the topological structure of this mapping.
In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for…
The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…
The bifurcation sets of polynomial functions have been studied by many mathematicians from various points of view. In particular, N\'emethi and Zaharia described them in terms of Newton polytopes. In this paper, we will show analogous…
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 $p$ be a prime number. Every two-variable polynomial $f(x_1, x_2)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to $\mathbb Z_p$. Our goal of this paper is…
This paper investigates the dynamical properties of Dickson polynomials over finite fields, focusing on the periodicity and structural behavior of their iterated sequences. We introduce and analyze the sequence $[D_n(x, \alpha) \mod (x^q -…
We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a…
We study the notion of Fagnano orbits for dual polygonal billiards. We used them to characterize regular polygons and we study the iteration of the developing map.
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…
There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus $g$ in positive characteristic $p$. In this chapter, I survey the current state of knowledge about these questions. I…
We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend…
We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the…
Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^\gamma (t^2-\|x\|^2)^{\mu-\frac12}$ on…
A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition…