光滑二元有理函数系数的自动渐近性
Automatic asymptotics for coefficients of smooth, bivariate rational functions
摘要：
本文考虑二元有理母函数F（x，y）= P（x，y）/ Q（x，y）= sum_{r, s} a_{r,s} x^r y^s，假定$Q$在其上为零的复代数曲线$\sing$是光滑的.系数a_{rs}的渐近公式由Pemantle和Wilson（2002）推导。这些公式是根据代数和拓扑不变量的极点簇，但到目前为止，这些不变量只能计算下的极小性假设，即优势鞍位于边界上的收敛域。在本文中，我们给出了一个有效的方法来计算拓扑不变量，从而渐近值a_{r,s}，没有极小性假设。这导致了理论上严格的算法，其实现正在http://www.mathemagix.org上进行。
英文摘要：
We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the coefficients a_{rs} were derived by Pemantle and Wilson (2002). These formulae are in terms of algebraic and topological invariants of the pole variety, but up to now these invariants could be computed only under a minimality hypothesis, namely that the dominant saddle lies on the boundary of the domain of convergence. In the present paper, we give an effective method for computing the topological invariants, and hence the asymptotics of the values a_{r,s}, without the minimality assumption. This leads to a theoretically rigorous algorithm, whose implementation is in progress at http://www.mathemagix.org .
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