Related papers: Counting Elements on Full Matrix Groups over Finit…

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number…

We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for…

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those…

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly…

Determinants of structured matrices play a fundamental role in both pure and applied mathematics, with wide-ranging applications in linear algebra, combinatorics, coding theory, and numerical analysis. In this work, the enumeration of…

An element $\alpha \in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $\alpha^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over…

In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…

By using the free field realizations, we analyze the representation theory of the W_{1+infinity} algebra with c=1. The eigenvectors for the Cartan subalgebra of W_{1+infinity} are parametrized by the Young diagrams, and explicitly written…

In this paper we present an algorithm for efficiently counting fixed points in a finite monoid $M$ under a conjugacy-like action. We then prove a formula for the character table of $M$ in terms of fixed points and radical, which allows for…

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient…

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof…

Let $\text{U}(n,\mathbb{F}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,\mathbb{F}_{q^2})$ which fixes a Hermitian form and $M\geq 2$ an integer. This is a companion paper to the previous works…

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…

We describe a way to approximate the matrix elements of a real power $\alpha$ of a positive (for $\alpha \ge 0$) or non-negative (for $\alpha \in \mathbb{R}$), infinite, bounded, sparse and Hermitian matrix $W$. The approximation uses only…

Let $\zeta$ be a fixed nonzero element in a finite field $\mathbb F_q$ with $q$ elements. In this article, we count the number of pairs $(A,B)$ of $n\times n$ matrices over $\mathbb F_q$ satisfying $AB=\zeta BA$ by giving a generating…

For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in…

In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at…