Section: New Results

Algebraic Diagonals and Walks

The diagonal of a multivariate power series $F$ is the univariate power series $\mathrm{𝖣𝗂𝖺𝗀}F$ 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. In  [28], Alin Bostan and Louis Dumont, together with Bruno Salvy (AriC), have studied 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 $\mathrm{𝖣𝗂𝖺𝗀}F$ is an algebraic function. They have proposed an algorithm for computing an annihilating polynomial of $\mathrm{𝖣𝗂𝖺𝗀}F$. They have given a precise bound on the size of this polynomial and show that generically, this polynomial is the minimal polynomial of $\mathrm{𝖣𝗂𝖺𝗀}F$ and that its size reaches the bound. Their algorithm runs in time quasi-linear in this bound, which grows exponentially with the degree of the input rational function. They have also addressed the related problem of enumerating directed lattice walks. The insight given by their study has led to a new method for expanding the generating power series of bridges, excursions and meanders. They have shown that their first $N$ terms can be computed in quasi-linear complexity in $N$, without first computing a very large polynomial equation. An extended version of this work has been presented in [4].