Small solutions of polynomial congruences
type: Item
instance of: scholarly article
title: Small solutions of polynomial congruences
zbMATH Open document ID: 1203.11002
author: M. Ram Murty
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2010-07-09T00:00:00Z
review text: Let \(\delta>0\) be fixed. If \(f(x)\in{\mathbb Z}[x]\) is a polynomial of degree \(\geq p^\delta\) and \(f(x)\equiv 0\pmod p\) has a solution, the author asks whether there is a solution \(x_0\) with \(|x_0|<p^{1-\varepsilon} \) for some \(\varepsilon=\varepsilon(\delta)\). For \(f(x)=x^q-a\), \(q\mid p-1\), which admits a solution modulo \(p\), he estimates the smallest solution \(x_0\) with \(0<x_0<p\) as \(x_0\ll p^{\frac32}q^{-1}\log p\) and by applying the Burgess character sum estimates [\textit{D. A. Burgess}, Proc. Lond. Math. Soc. (3) 13, 524--536 (1963; Zbl 0123.04404)], and estimates of certain exponential sums due to \textit{J. Bourgain, A. A. Glibichuk} and \textit{J. V. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)] derives refinements of this result.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 5734646
zbMATH Keywords: polynomial congruences
MaRDI profile type: Publication
full work available at URL: s13226-010-0015-z
OpenAlex ID: W1998375233
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DOI: 10.1007/S13226-010-0015-Z
description: scientific article; zbMATH DE number 5734646