On the reduction modulo \(p\) of absolutely irreducible polynomials
type: Item
instance of: scholarly article
title: On the reduction modulo \(p\) of absolutely irreducible polynomials
zbMATH Open document ID: 0942.11033
published in: Monatshefte für Mathematik
publication date: 2000-08-20T00:00:00Z
review text: Let \(K\) be an algebraic extension of \(\mathbb Q\) of degree \(d\), and let \(F(X,Y)\in K[X,Y]\) be an absolutely irreducible polynomial. We know that there exists a positive integer \(A(F)\) such that, for any prime ideal \(\wp\) of the ring of integer \(\mathcal O_K\) of \(K\) with norm \(\geq A(F)\), the reduction of \(F\) modulo \(\wp\) is absolutely irreducible. For \(K=\mathbb Q\), this is an old result of Ostrowski. In this paper, the author gives an explicit value for \(A(F)\) which is of the same type as the ones obtained by \textit{W. Ruppert} [J. Reine Angew. Math. 369, 167-191 (1986; Zbl 0584.14012)] and by \textit{U. Zannier} [Arch. Math. 68, 129-138 (1997)]. Explicitly, assume that \(F\) has coefficients in \(\mathcal O_K\) and denote by \(N\) the maximum of \(\deg_X(F)\) and \(2g+\deg_Y(F)+1\) where \(g\) is the genus of the curve \(F(X,Y)=0\). Then, one can take \[ A(F) = H(F)^{64d^3 N^9} N^{97 d^3 N^{10}}, \] where \(H(F)\) denotes the absolute Weil height of the vector formed by 1 and the coefficients of \(F\). The proof of this result combines an explicit version of the Riemann-Roch theorem due to \textit{W. Schmidt} [J. Number Theory 39, 181-224 (1991; Zbl 0764.11046)] together with an explicit version of the implicit function theorem due to \textit{M. Laurent} and the reviewer [Ann. Inst. Fourier 49, 27-55 (1999; Zbl 0923.11105)].
reviewed by: Damien Roy
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 13P05
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 1423220
zbMATH Keywords: polynomials
zbMATH Keywords: curves
zbMATH Keywords: irreducibility
zbMATH Keywords: reduction modulo \(p\)
zbMATH Keywords: height
zbMATH Keywords: explicit bounds
author: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: s006050050013
OpenAlex ID: W2322067139
DOI: 10.1007/S006050050013
description: scientific article; zbMATH DE number 1423220