Related papers: On the distribution of reducible polynomials

An irreducible representation of a reductive Lie algebra, when restricted to a Cartan subalgebra, decomposes into weights with multiplicity. The first part of this paper outlines a procedure to compute symmetric polynomials (e.g., power…

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and…

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include…

Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log n}$, the degree…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

For each integer $t>0$ and each complex simple Lie algebra $\mathfrak{g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak{g}$ whose highest weight has height $t$. As a corollary, we classify…

In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular,…

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an…

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

We answer an open question in the theory of transducer degrees initially posed in [3], on the structure of polynomial transducer degrees, in particular the question of what degrees, if any, lie below the degree of $n^3$. Transducer degrees…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and…

A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x^n in R[x], where (R, m) is an Artinian local ring with m^2 = 0.

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the…

We investigate the analogues, in $\mathbb{F}_q[t]$, of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too…

We develop a method of reducing the size of quantum minors in the algebra of n x n quantum matrices. The method is used to show that quantum determinantal factor rings of n x n quantum matrices over the complex numbers are maximal orders,…

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We…