相关论文: On the reducibility type of trinomials

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…

In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We…

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

As an extension of the classical irreducibility result of Dumas, a factorization result for polynomials over any valued field with a Krull valuation of arbitrary rank is proved. Further, a lower degree factor bound on factors of a given…

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…

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of…

An abelian monogenic polynomial $f(x)\in {\mathbb Z}[x]$ is a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$, such that the Galois group of $f(x)$ over ${\mathbb Q}$ is abelian, and…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We…

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em…

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…

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$)…

This article addresses the existence of $\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $\P^N$ where each component is a degree 2 homogeneous form in $N+1$…

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the…

We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the…

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence…

Let Y_n(t) denote the set of all polynomials over the ring Z which are reducible over the field Q and of degree n>1 and of height not greater than t. We show that the true order of magnitude of |Y_n(t)| equals t^2 log t in the special case…

A polynomial with rational coefficients is said to be pure with respect to a rational prime $p$ if its Newton polygon has one slope. In this article, we prove that the number of irreducible factors of the $n$-th iterate of a pure polynomial…

We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f.…