On the complexity of computing integral bases of function fields

Simon Abelard (1, 2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 David R. Cheriton School of Computer Science

Let $\mathcal{C}$ be a plane curve given by an equation $f(x, y) = 0$ with $f ∈ K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to Böhm et al. and achieve a quasi-optimal runtime.

Mots clés: Puiseux series; Linear algebra; Polynomial matrices

Domaines: Calcul formel [cs.SC]; Géométrie algébrique [math.AG]; Complexité [cs.CC]; Algèbre commutative [math.AC]

Simon Abelard. On the complexity of computing integral bases of function fields. CASC 2020 - Computer Algebra in Scientific Computing, Sep 2020, Linz / Virtual, Austria. pp.42-62, ⟨10.1007/978-3-030-60026-6_3⟩. ⟨hal-02568086v2⟩