Related papers: Exponential sums with reducible polynomials

Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has the same…

The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove…

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

We study the number of irreducible polynomials over $\mathbf{F}_{q}$ with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree $n$ with $r$ coefficients…

In this paper we study the asymptotic zero distribution of eigenpolynomials for degenerate exactly-solvable operators. We present an explicit conjecture and partial results on the growth of the largest modulus of the roots of the unique and…

We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of…

We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$…

This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are…

In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $\mu_j,$ then $A$…

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a…

Let $F(X_1,X_2)\in\mathbb{Z}[X_1,X_2] $ be an irreducible binary form of degree $3$ and $h$ an arithmetic function. We give some estimates for the average order $\sum_{\substack{|n_1|\leq x,|n_2|\leq x}}h(F(n_1,n_2))$ when $h$ satisfy…

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq…

A detailed analysis of the remainder obtained by truncating the Euler series up to the $n$th-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse…

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the…

We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_n^{(n)}(z)$. The asymptotic expansions of the scaled polynomials $B_n^{(n)}(nz)$ are obtained as $n\to\infty$ when…

We consider the unconstrained optimization of multivariate trigonometric polynomials by the sum-of-squares hierarchy of lower bounds. We first show a convergence rate of $O(1/s^2)$ for the relaxation with degree $s$ without any assumption…

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group…

In a recent paper \cite{Jan}, Jankauskas proved some interesting results concerning the reducibility of quadrinomials of the form $f(4,x)$, where $f(a,x)=x^{n}+x^{m}+x^{k}+a$. He also obtained some examples of reducible quadrinomials…