Section:
New Results
Algebraic Diagonals and Walks
The diagonal of a multivariate power series is the univariate power series
generated by the diagonal terms of . Diagonals form an
important class of power series; they occur frequently in number theory,
theoretical physics and enumerative combinatorics.
In [7] we study algorithmic questions related to
diagonals in the case where is the Taylor expansion of a bivariate
rational function. It is classical that in this case is an
algebraic function. We propose an algorithm that computes an annihilating
polynomial for . 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 terms can be computed in quasi-linear complexity in
, without first computing a very large polynomial equation. An extended
version of this work is presented in [13] .