代数对角线与漫步

A. Bostan, L. Dumont, B. Salvy
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引用次数: 5

摘要

多元幂级数$F$的对角线是由F的对角线项生成的单变量幂级数DiagF。对角线是幂级数的一个重要类别;它们经常出现在数论、理论物理和计数组合学中。我们研究了关于对角线的算法问题,其中F是二元有理函数的泰勒展开式。这是经典的,在这种情况下,DiagF是一个代数函数。我们提出了一种计算DiagF的湮灭多项式的算法。一般来说,它是它的最小多项式,在时间上是拟线性的。我们证明了这个最小多项式对于输入有理函数的阶有一个指数大小。然后我们讨论了枚举有向格行走的相关问题。我们的研究为扩大桥梁、短途和蜿蜒的发电系列提供了一种新的方法。我们证明了它们的前N项可以在N的拟线性复杂度下计算,而不需要首先计算一个非常大的多项式方程。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Algebraic Diagonals and Walks
The diagonal of a multivariate power series $F$ is the univariate power series DiagF generated by the diagonal terms of F. Diagonals form an important class of power series; they occur frequently in number theory, theoretical physics and enumerative combinatorics. We study algorithmic questions related to diagonals in the case where F is the Taylor expansion of a bivariate rational function. It is classical that in this case DiagF is an algebraic function. We propose an algorithm that computes an annihilating polynomial for DiagF. Generically, it is its minimal polynomial and is obtained in time quasi-linear in its size. We show that this minimal polynomial has an exponential size with respect to the degree of the input rational function. We then address the related problem of enumerating directed lattice walks. The insight given by our study leads to a new method for expanding the generating power series of bridges, excursions and meanders. We show that their first N terms can be computed in quasi-linear complexity in N, without first computing a very large polynomial equation.
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