On the reduction modulo \(p\) of absolutely irreducible polynomials
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)].
scientific article; zbMATH DE number 1423220