Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ 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 B\"ohm et al. and achieve a quasi-optimal runtime.

中文翻译：

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论函数域积分基的计算复杂度

令 $\mathcal{C}$ 是由方程 $f(x,y)=0$ 给出的平面曲线，其中 $f\in K[x][y]$ 是一个无二阶平方多项式。我们研究了计算代数函数域 $K(\mathcal{C})$ 的积分基的问题，并为处理这个问题的三个已知算法给出了新的复杂度界限。对于每个算法，我们研究其子程序，并在可能的情况下修改或替换它们以利用更快的原语。然后，我们结合复杂度结果来推导出每个算法的整体复杂度估计。特别是，由于 B\"ohm 等人，我们修改了算法并实现了准最优运行时间。