Some number-theoretic combinatorial problems
type: Item
instance of: scholarly article
title: Some number-theoretic combinatorial problems
zbMATH Open document ID: 1263.11113
DOI: 10.1134/S0001434609090107
author: V. K. Leont'ev
published in: Mathematical Notes
publication date: 2010-01-12T00:00:00Z
review text: From the text: ``The author considers a number of number-theoretic combinatorial problems arising in the study of problems of graph theory and of the theory of error-correcting codes. He obtain simple expressions for a number of generating functions and, from them, derives consequences required for solving original problems. Suppose that \(\mathbb F_q\) is a finite field of \(q\) elements and \(\mathbb F_q[x]\) is the ring of polynomials over the field \(\mathbb F_q\). Further, we only consider normalized polynomials in \(\mathbb F_q[x]\). The Möbius function \(\mu(g)\) given on the elements of the ring \(\mathbb F_q[x]\), is defined by the canonical method as \[ \mu(g)=\begin{cases} 1 &\text{ for } g=1,\\ (-1)^r &\text{ for } g=v_1v_2\cdots v_r,\\ 0 &\text{ if } g \text{ is divisible by a square}.\end{cases} \tag{1} \] Here the \(v_i\) are irreducible polynomials of the factorial ring \(\mathbb F_q[x]\). The Möbius function (1) is a direct analog of the classical number-theoretic Möbius function defined in the ring of integers. Consider the following generating function: \(F_q(z)=\sum_{g\in\mathbb F_q[x]} \mu(g)z^{\deg g}\). Theorem 1. The following relation holds: \(F_q(z)=1-qz\). '' A number of further interesting results are proved.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 5656338
zbMATH Keywords: generating function
zbMATH Keywords: finite field
zbMATH Keywords: Möbius function
zbMATH Keywords: polynomial ring over \(F_q\)
zbMATH Keywords: irreducible polynomial
zbMATH Keywords: Hamiltonian cycle
zbMATH Keywords: Euler function
MaRDI profile type: Publication
cites work: New constructions of optimal cyclically permutable constant weight codes
cites work: When the cartesian product of directed cycles is Hamiltonian
cites work: Q3939804
full work available at URL: s0001434609090107
OpenAlex ID: W2029727862
description: scientific article; zbMATH DE number 5656338