Algebraic diagonals and walks: Algorithms, bounds, complexity

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 relate...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 83; pp. 68 - 92
Main Authors: Bostan, Alin, Dumont, Louis, Salvy, Bruno
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.11.2017
Elsevier
Series:Special issue on the conference ISSAC 2015: Symbolic computation and computer algebra
Subjects:
Algorithms
Computer Science
Diagonals
Symbolic Computation
Walks
68W30
33F10
Algorithms
05A15
Walks
Diagonals
walks
algorithms
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary: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. We give a precise bound on the size of this polynomial and show that, generically, this polynomial is the minimal polynomial and that its size reaches the bound. The algorithm runs in time quasi-linear in this bound, which grows exponentially with 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.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2016.11.006