A note on simultaneous congruences
type: Item
instance of: scholarly article
title: A note on simultaneous congruences
zbMATH Open document ID: 0852.11017
published in: Journal of Number Theory
publication date: 1996-11-28T00:00:00Z
review text: Let \(f_1, \dots, f_d\) be polynomials in \(\mathbb{Z} [x_1, \dots, x_d]\) with respective degrees \(k_1, \dots, k_d\), and write \[ J({\mathbf f}, {\mathbf x})= \text{det} \biggl( {{\partial f_j} \over {\partial x_i}} ({\mathbf x}) \biggr)_{1\leq i,j\leq d}. \] When \(p\) is a prime number and \(s\) is a natural number, let \({\mathcal N} ({\mathbf f}; p^s)\) denote the number of solutions of the simultaneous congruences \(f_j (x_1, \dots, x_d) \equiv 0\pmod {p^s}\), with \(1\leq x_i\leq p^s\) \((1\leq i, j\leq d)\) and \((J({\mathbf f}; {\mathbf x}),p)=1\). In this paper the author uses an elementary argument to prove that \[ {\mathcal N}({\mathbf f}; p^s)\leq k_1 \cdots k_d. \] The result is best possible, in view of the example \(f_j ({\mathbf x})= x_j^{kj} -1\) for \(1\leq j\leq d\). The author also provides a sketch of an alternative proof of the above result by using Bézout's theorem.
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 894608
zbMATH Keywords: polynomials in many variables
zbMATH Keywords: Waring's problem
zbMATH Keywords: simultaneous congruences
author: Trevor D. Wooley
reviewed by: Ming-Chit Liu
MaRDI profile type: Publication
full work available at URL: jnth.1996.0078
OpenAlex ID: W2013855456
DOI: 10.1006/JNTH.1996.0078
description: scientific article; zbMATH DE number 894608