On the reduction modulo \(p\) of plane algebraic curves (Q1592795)

If \(F(X_1,\dots,X_n)\) is an absolutely irreducible polynomial with integer coefficients, then it is known since \textit{A. Ostrowski} [Nachr. K. Ges. Wiss. Göttingen 1919, 279-298 (1919; JFM 47.0067.04)] that the reduction of \(F\) modulo \(p\) remains irreducible for all prime numbers \(p\geq \Omega(F)\). Moreover, \(\Omega(F)\) can be bounded by a function of polynomial growth in the size of \(F\) [see \textit{U. Zannier}, Arch. Math. 68, 129-138 (1997; Zbl 0977.11011), \textit{W. Ruppert}, J. Number Theory 77, 62-70 (1999; Zbl 0931.11005) and \textit{D. Poulakis}, Monatsh. Math. 129, No. 2, 139-145 (2000; Zbl 0942.11033)]. Let \(F(X,Y,Z)\) be an absolutely irreducible polynomial with coefficients in a number field \(K\), and assume that \(F(X,Y,Z)=0\) defines a non-singular curve. The author proves that, for all primes \(\wp\) in the ring of integers of \(K\) whose norm is \(\geq \Omega(F)\), the reduction modulo \(\wp\) of the curve \(F(X,Y,Z)=0\) remains non-singular. The bound for \(\Omega(F)\) is again of polynomial type. The proof uses a theorem of Lazard on elimination [see \textit{D. Lazard}, Bull. Soc. Math. Fr. 105, 165-190 (1977; Zbl 0447.13008)].