The invariant polynomials degrees of the Kronecker sum of two linear operators and additive theory
type: Item
instance of: scholarly article
title: The invariant polynomials degrees of the Kronecker sum of two linear operators and additive theory
zbMATH Open document ID: 0966.15011
DOI: 10.1016/S0024-3795(00)00125-7
author: Cristina Caldeira
author: J. A. Dias Da Silva
published in: Linear Algebra and its Applications
publication date: 2001-08-16T00:00:00Z
review text: Let \(A\), \(B\) be finite nonempty subsets of a field \(\mathbb{F}\). Write \(A+ B=\{a+b: a\in A, b\in B\}\) and denote by \(\mu_i= \mu_i(A,B)\) \((i= 1,2,\dots)\) the number of elements of \(A+B\) that can be represented in the form \(a+b\) in at least \(i\) ways. Let \(p\) (\(=\infty\) or a prime) be the additive order of the nonzero elements of \(\mathbb{F}\). The authors prove that \[ \sum^\ell_{i=1} \mu_i\geq \ell\min\{p,|A|+|B|- \ell\} (\ell= 1,\dots, \min\{|A|,|B|\}). \] This gives the Cauchy-Davenport Theorem: \(|A+B|\geq \min\{p,|A|+|B|- 1\}\), when \(\ell= 1\) and \(\mathbb{F}= \mathbb{F}_p\). The full result over \(\mathbb{F}_p\) was proved by \textit{J. M. Pollard} [J. London Math. Soc., II. Ser. 8, 460-462 (1974; Zbl 0322.10024)]. The proof, which is quite different from Pollard's, rests on the following observation: let \(V\), \(W\) be vector spaces of dimensions \(|A|\), \(|B|\) and let \(f\), \(g\) be linear transformations on \(V\), \(W\) with spectra \(A\), \(B\); then the spectrum of the Kronecker sum \(f\otimes I_W+ I_V\otimes g\) is the family \((a+b)_{(a,b)\in A\times B}\). Quite detailed calculations with invariant factors are required in the course of the proof.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 1523265
zbMATH Keywords: additive theory
zbMATH Keywords: Cauchy-Davenport Theorem
zbMATH Keywords: Kronecker sum
zbMATH Keywords: invariant factors
reviewed by: G. E. Wall
MaRDI profile type: Publication
cites work: On the Addition of Residue Classes
cites work: A Historical Note
cites work: A note on the minimal polynomial of the Kronecker sum of two linear operators
cites work: Q4895030
cites work: A Generalisation of the Theorem of Cauchy and Davenport
cites work: Matrices with prescribed rows and invariant factors
cites work: Interlacing inequalities and control theory
cites work: Controllability and Hermite indices of matrix pairs
full work available at URL: s0024-3795(00)00125-7
OpenAlex ID: W1971880286
Recommended article: Q4832292
Recommended article: Bemerkungen zur additiven Zahlentheorie. I. Mittlere Ordnung.
Recommended article: Q4668899
Recommended article: Q5722324
Recommended article: Q3974937
Recommended article: A Cauchy-Davenport theorem for linear maps
Recommended article: Symmetric function means and permanents
Recommended article: Additive and Multiplicative Structure in Matrix Spaces
Recommended article: The number of distinct subset sums of a finite set of vectors
Recommended article: Regular $h$-ranges and weakly pleasant $h$-bases.
description: scientific article; zbMATH DE number 1523265