Q873204
type: Item
instance of: scholarly article
title: Andrzej Schinzel selecta. Volume I: Diophantine problems and polynomials. Volume II: Elementary, analytic and geometric number theory. Edited by Henryk Iwaniec, Władysław Narkiewicz and Jerzy Urbanowicz
zbMATH Open document ID: 1115.11002
DOI: 10.4171/038
author: Andrzej Schinzel
published in: Heritage of European Mathematics
publication date: 2007-03-29T00:00:00Z
review text: Nobody is able to present seriously all the works contained in these two volumes except A. Schinzel himself. A. Schinzel wrote his first paper at the age of 17: ``\textit{Sur la décomposition des nombres naturels en somme de nombres triangulaires distincts}'', Bull. Acad. Polon. Sci., 1954. Most of his first papers were much influenced by his supervisor W. Sierpiński. The central theme of Schinzel's work is arithmetical and algebraic properties of polynomials in one or several variables, in particular questions of irreducibility and zeros of polynomials; this concerns about one third of his papers and he wrote two books on this subject, each containing a lot of old and new information. The selection presented in these two volumes contains 100 papers chosen among more than 200 papers published by Schinzel, and also a list of unsolved problems and unproved conjectures proposed by Schinzel in the years 1956--2006. This collection is organized into 13 sections, each theme being presented and commented by an expert. I present this list below and for each section I select one result of Schinzel. My main criterion is simplicity and elegance and the second one is novelty and originality, many of these results were in advance when they appeared. But I note that most of the deepest results of Schinzel are very long to state, they contain long lists of explicity special cases and it was impossible to reproduce them here. A. ``Diophantine equations and integral forms'', commented by R. Tijdeman -- 10 papers. Reference: ``On the equation \(y^m=P(x)\)'' [with \textit{R. Tijdeman}], Acta Arith. \ (1976). -- If a polynomial \(P(x)\) with rational coefficients has at least two distinct zeros then the equation \(\,y^m=P(x)\), where \(x\) and \(y\) are integers, \(| y| >1\), implies \(\,m<c(P)\) (effective). B. ``Continued fractions and integral forms'', commented by E. Dubois -- 3 papers. Reference: ``On some problems of the arithmetical theory of continued fractions'', Acta Arith. \…
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 01A75
Mathematics Subject Classification ID: 12-03
zbMATH DE Number: 5138337
reviewed by: Maurice Mignotte
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/038
OpenAlex ID: W1544032248
Wikidata QID: Q56814138
label: Andrzej Schinzel selecta. Volume I: Diophantine problems and polynomials. Volume II: Elementary, analytic and geometric number theory. Edited by Henryk Iwaniec, Władysław Narkiewicz and Jerzy Urbanowicz
description: scientific article; zbMATH DE number 5138337