Single-exponential bounds for diagonals of D-finite power series

TL;DR

This paper proves that diagonals of D-finite power series are D-finite, fixes a gap in Gessel's proof, and provides explicit exponential bounds on their defining differential equations.

Contribution

It offers a new proof of Gessel's result on diagonals of D-finite series and extends it to all D-finite power series, with explicit bounds.

Findings

Diagonals of D-finite series are D-finite.

Provided a corrected proof of Gessel's theorem.

Established exponential bounds on differential equations.

Abstract

D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.