On the distribution of solutions to polynomial congruences
type: Item
instance of: scholarly article
title: On the distribution of solutions to polynomial congruences
zbMATH Open document ID: 1334.11018
author: Igor E. Shparlinski
published in: Archiv der Mathematik
publication date: 2013-01-14T00:00:00Z
review text: After having defined a system of congruences modulo a prime \(p\) (in the family \(\mathfrak F_{m,n}\) of \(n\) polynomials in \(m \geq n+1\) variables, with integer coefficients) and its set of solutions \(\chi_p\), the paper clarifies the properties of a well-shaped domain \(\Omega \in [0,1]^m\) and denotes by \(T_p(\Omega)\) the number of points falling inside of \(\Omega\). Then the paper establishes the following asymptotic formula for the number of solutions in general domains: \[ T_p(\Omega) = \# \chi_p (\mu (\Omega) + O(p^{-1/2(n+1)} \log p)), \] where \( \# \chi_p = T_p([0,1]^m)\) and \(\mu\) is the Lebesgue measure on \([0,1]^m\). It extends the result of \textit{É. Fouvry} [Isr. J. Math. 120, Part A, 81--96 (2000; Zbl 1010.11045)] valid in small cubic boxes only. The proof employs a combinatorial approximation of \(\Omega \) supplied by \textit{W. M. Schmidt} [Acta Arith. 27, 385--396 (1975; Zbl 0274.10038)], an equation about well-shaped domains given by \textit{H. Weyl} [Am. J. Math. 61, 461--472 (1939; Zbl 0021.35503)] and a theorem on the number of points of varieties in finite fields provided by \textit{S. Lang} and \textit{A. Weil} [Am. J. Math. 76, 819--827 (1954; Zbl 0058.27202)]. The author remarks the effectiveness of his approach, non-standard with respect to others (e.g., by \textit{A. Granville} et al. [J. Number Theory 112, No. 2, 216--237 (2005; Zbl 1068.11043)]), and, on the basis of an achievement from \textit{A. Ayyad} et al. [J. Number Theory 59, No. 2, 398--413 (1996; Zbl 0869.11003)], he suggests a combination with an estimate found by the same \textit{I. E. Shparlinski} [Proc. Japan Acad., Ser. A 83, No. 2, 5--9 (2007; Zbl 1123.11026)]. In order to enrich the paper's argumentations, the author recalls a vast literature dealing with the number and distribution of solutions for systems of congruences like, e.g., by \textit{E. Fouvry} and \textit{N. Katz} [J. Reine Angew. Math. 540, 115--166 (2001; Zbl 0986.11054)], by \textit{Wenzhi Lu…
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 6125029
zbMATH Keywords: multivariate congruences
zbMATH Keywords: polynomials
zbMATH Keywords: distribution of points
zbMATH Keywords: well-shaped domains
MaRDI profile type: Publication
full work available at URL: s00013-012-0436-5
OpenAlex ID: W2081858067
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DOI: 10.1007/S00013-012-0436-5
description: scientific article; zbMATH DE number 6125029