arXiv subjects

Natalio H. Guersenzvaig

Publications and source records attributed to Natalio H. Guersenzvaig.

6 recordsLinked to original sources

arXiv2014

Generalized Artin-Schreier polynomials

Let $F$ be a field of prime characteristic $p$ containing $F_{p^n}$ as a subfield. We refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. Suppose that $q(X)$ is irreducible and let $C_{q(X)}$ be the companion matrix of $q(X)$. Then $ad\, C_{q(X)}$ has such highly unusual properties that any $A\in{\mathfrak{ gl}}(m)$ such that $ad\, A$ has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin-Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a 1-dimensional Lie algebra over a field of characteristic $p$, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin-Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.

math.RA↗

arXiv2013

Elementary criteria for irreducibility of $f(X^n)$

Very simple sufficient conditions for the irreducibility of $f(X^n)$ over an arbitrary unique factorization domain $Z$ are established via a generalization of a well known theorem of A. Capelli.

math.RA↗

arXiv2013

Roots multiplicity and square-free factorization of polynomials using companion matrices

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $α$ of $f$ in the algebraic closure of $F$, $M_f(α)$ is equal to the multiplicity $m(α)$ of $α$ as a root of $f$. As an application of $M_f$ we give a new method to compute in $F[X]$ each component of the square-free factorization $f=P_1P_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-α$ with $m(α)=k$, for $k=1, \dots, m=\max m(α)$.

math.RA↗

arXiv2013

Is every matrix similar to a polynomial in a companion matrix?

Given a field $F$, an integer $n\geq 1$, and a matrix $A\in M_n(F)$, are there polynomials $f,g\in F[X]$, with $f$ monic of degree $n$, such that $A$ is similar to $g(C_f)$, where $C_f$ is the companion matrix of $f$? For infinite fields the answer is easily seen to positive, so we concentrate on finite fields. In this case we give an affirmative answer, provided $|F|\geq n-2$. Moreover, for any finite field $F$, with $|F|=m$, we construct a matrix $A\in M_{m+3}(F)$ that is not similar to any matrix of the form $g(C_f)$. Of use above, but also of independent interest, is a constructive procedure to determine the similarity type of any given matrix $g(C_f)$ purely in terms of $f$ and $g$, without resorting to polynomial roots in $F$ or in any extension thereof. This, in turn, yields an algorithm that, given $g$ and the invariant factors of any $A$, returns the elementary divisors of $g(A)$. It is a rational procedure, as opposed to the classical method that uses the Jordan decomposition of $A$ to find that of $g(A)$. Finally, extending prior results by the authors, we show that for an integrally closed ring $R$ with field of fractions $F$ and companion matrices $C,D$ the subalgebra $R< C,D>$ of $M_n(R)$ is a free $R$-module of rank $n+(n-m)(n-1)$, where $m$ is the degree of $\gcd (f,g)\in F[X]$, and a presentation for $R< C,D>$ is given in terms of $C$ and $D$. A counterexample is furnished to show that $R< C,D>$ need not be a free $R$-module if $R$ is not integrally closed. The preceding information is used to study $M_n(R)$, and others, as $R[X]$-modules.

math.RA↗

arXiv2010

A Closed Formula for the Product in Simple Integral Extensions

Let $ξ$ be an algebraic number and let $α,β\in \mathbb Q[ξ]$. An explicit formula for the coordinates of the product $αβ$ is given in terms of the coordinates of $α$ and $β$ and the companion matrix of the minimal polynomial of $ξ$. The formula as well as its proof extend to fairly general simple integral extensions.

math.RA↗

arXiv2010

Subalgebras of Matrix Algebras Generated by Companion Matrices

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R< C,D>=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given.

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