On the computation of the conductor of an affine algebraic curve (Q1814214)

Let \(C=\text{Spec} R\) be an affine reduced curve over a field \(k\). In this paper we want to compute the conductor \(I\) of \(R\) in its normalization \(\overline R\). In particular if \(C\) has only ordinary generic singularities we show that there is an algorithm to carry out this computation. If \(k=\mathbb{Z}_ p\) this algorithm can be easily implemented on personal computers using common languages like Basic, C, APL. This has been done by the second author. If \(k=\mathbb{Q}\), for the same goal, one can use computer algebra systems like Macsyma, Reduce, Maple. These programs allow to construct computations of the conductor over any field \(k\) (recall that any field contains \(\mathbb{Z}_ p\) or \(\mathbb{Q})\). In the course of the paper we extend also various theoretical results of the first named author [J. Lond. Math. Soc., II. Ser. 24, 85-96 (1981; Zbl 0492.14017)] and both authors [Manuscr. Math. 68, No. 1, 1-7 (1990; Zbl 0709.13010)].