Algebraic Diagonals and Walks

TL;DR

This paper presents an efficient algorithm for computing algebraic diagonals of bivariate rational functions and applies it to enumerate lattice walks, enabling rapid calculation of generating series terms.

Contribution

It introduces a quasi-linear time algorithm to find minimal polynomials of diagonals and applies this to improve enumeration of lattice walks.

Findings

Algorithm computes minimal polynomial of diagonal in quasi-linear time.

Minimal polynomial size can be exponential in input degree.

New method efficiently generates initial terms of lattice walk series.

Abstract

The diagonal of a multivariate power series F is the univariate power series Diag(F) 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 Diag(F) is an algebraic function. We propose an algorithm that computes an annihilating polynomial for Diag(F). 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.