Polynomials determined by a few of their coefficients.
type: Item
instance of: scholarly article
title: Polynomials determined by a few of their coefficients.
zbMATH Open document ID: 1090.11018
DOI: 10.1016/S0019-3577(04)90015-8
author: Lajos Hajdu
author: Andrzej Schinzel
author: Kálmán Győry
author: Ákos Pintér
published in: Indagationes Mathematicae. New Series
publication date: 2005-01-03T00:00:00Z
review text: Let \(K\) be a field of characteristic zero. The authors consider the following Problem: Is it true that a monic polynomial \(f\in k[x]\) of degree \(n\) with exactly \(k\) distinct zeros is determined up to finitely many possibilities by any \(k\) of its non-zero proper coefficients? Here a coefficient is called proper if it is the coefficient of \(x^i\) for \(i<n\) and the assignment of \(k\) non-zero proper coefficients is to be meant as the assigment of \(k\) pairs \((i,a_i)\) where \(0\leq i<n\) and \(0\neq a_i\in K\). The answer is positive in a number of cases. To give an example, Theorem 1 of the paper states that if \(k=2\) the polynomial is determined up to \(n(n-1)[n/2]\) possibilities. Restricting to the case \(K={\mathbb Q}\), the authors prove as an application the finiteness, under suitable conditions, of the solutions \((x,y)\in {\mathbb Z}^2\) of the superelliptic Diophantine equation \(f(x)+g(x) = by^l\), where \(f,g\in{\mathbb Q}[x]\), \(b\in {\mathbb Q}\setminus \{0\}\) and \(l\) is an integer \(\geq 2\).
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 2124639
zbMATH Keywords: polynomials
zbMATH Keywords: monic polynomial
zbMATH Keywords: proper coefficients
zbMATH Keywords: superelliptic Diophantine equation
MaRDI profile type: Publication
cites work: On \(S\)-integral solutions of the equation \(y^ m=f(x)\)
cites work: Q4432060
cites work: Q4325515
cites work: Q4293472
cites work: On sums of roots of unity (Solution of two problems of R. M. Robinson)
cites work: Q4428509
cites work: On the Diophantine equation \(1^k+2^k+\dots +x^k+R(x)=y^2\)
full work available at URL: s0019-3577(04)90015-8
OpenAlex ID: W1978189754
description: scientific article; zbMATH DE number 2124639