Related papers: Exceptional Scattered Polynomials

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q,…

In this paper, we compute the size of the exceptional set in a generalized Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$ with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0 \leq i, j…

We identify all non-splitting bi-unitary perfect polynomials over the field $\mathbb{F}_4$, which admit at most four irreducible divisors. There is an infinite number of such divisors.

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K= \mathbb{Q}(\theta)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted…

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in…

A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…

Let $n > k > t \geq j \geq 1$ be integers. Let $X$ be an $n$-element set, ${X\choose k}$ the collection of its $k$-subsets. A family $\mathcal F \subset {X\choose k}$ is called $t$-intersecting if $|F \cap F'| \geq t$ for all $F, F' \in…

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact…

We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category which does not have a full exceptional collection consisting of line…

We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern…

Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate…

Let $\mathbb{F}_q$ be a finite field with $q$ elements. M. Gerstenhaber and Irving Reiner has given two different methods to show the number of matrices with a given characteristic polynomial. In this talk, we will give another proof for…

The maximum gap $g(f)$ of a polynomial $f$ is the maximum of the differences (gaps) between two consecutive exponents that appear in $f$. Let $\Phi_{n}$ and $\Psi_{n}$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial,…

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable;…

Let $\alpha(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in the $p$-random subset of $\mathbb{F}_q^d$. In this note, we determine the order of magnitude of $\alpha(\mathbb{F}_q^{3},p)$ up to a…

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…

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and…

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of $f(x)$ in each of these families having: squarefree discriminant;…

We study the exceptional set estimate for projections in $\mathbb{F}_q^n$. For each $V\in G(k,\mathbb{F}^n_q)$, let $$ \pi_V: \mathbb{F}_q^n\rightarrow V $$ be the projection map. We prove the following result: If $A\subset \mathbb{F}_q^n$…