Complexity of computation of embedded resolution of algebraic curves

Abstract

We study the complexity of an algorithm we gave in a former paper to compute an embedded resolution of an irreducible singular algebraic curve.

This is more complex than just finding the Puiseux expansion associated to the curve, but this computation is also more interesting because it gives not only the Puiseux pairs, and the Puiseux series but also a way to work on some other invariants of the curve (see [D] for a study of complexity of computing Puiseux Pairs, with an emphasis on the reducible case, and algebraic numbers, and [5D] for the related algorithms).

For example it will allow the mathematician to work on the mixed Hodge structure of the curve.

This complexity is shown to be polynomial in terms of the degree d of the polynomial of the curve.

We will try to make a study of the complexity strongly related to the real algorithm we are using; in fact this study comes after, and is motivated by, two implementations we made of resolutions of irreducible curves ([H.M.])*.

Unité associée au CNRS n^o 169 and GRECO Calcul Formel n^o 60