Related papers: Generalized Wall-Sun-Sun primes and monogenic powe…

We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the…

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo…

In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property ''power stable range one'' if for all a, b $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $\lambda$…

We study a function field version of a classical problem concerning square-free values of polynomials evaluated at primes. We show that for a square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density as $n\to \infty$ of…

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…

Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes…

Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those…

The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the…

Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\mathbb{F}_q(t)$, where $t$ is transcendental over $\mathbb{F}_q$. We prove that when $n$ is a…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound…

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.

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We show…

Given $f \in \mathbb{Z}[x]$ and $n \in \mathbb{Z^{+}}$, the $\emph{discriminator}$ $D_f(n)$ is the smallest positive integer $m$ such that $f(1), \ldots, f(n)$ are distinct mod $m$. In a recent paper, Z.-W. Sun proved that $D_f(n) =…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

Given a power $q$ of a prime number $p$ and "nice" polynomials $f_1,...,f_r\in\bbF_q[T,X]$ with $r=1$ if $p=2$, we establish an asymptotic formula for the number of pairs $(a_1,a_2)\in\bbF_q^2$ such that…

Let $(\mathcal{A}_i)_{i \in [s]}$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two…

Let $A = \mathbb{F}_p$ or $\mathbb{Z}_p$, and let $R = A[[x_1]][[x_2; \sigma_2, \delta_2]]\dots[[x_n;\sigma_n,\delta_n]]$, an iterated local skew power series ring over $A$. Under mild conditions, we show that (multiplicative) monomial…