Single-exponential bounds for diagonals of D-finite power series
Shaoshi Chen, Frédéric Chyzak, Pingchuan Ma, Chaochao Zhu
TL;DR
This work proves that diagonals of multivariate $D$-finite power series remain $D$-finite and provides quantitative bounds on the differential operators annihilating these diagonals. It advances two main approaches: a gap-fixed reinterpretation of Gessel's diagonal proof and Lipshitz's diagonal-bounding framework, culminating in explicit single-exponential bounds for the degree and order of the annihilating operators in terms of the input system. For the primary diagonal, bounds are polynomial-exponential in the input data, while the complete diagonal admits either a doubly exponential bound via iteration or a single-exponential bound via a one-step construction, depending on the method chosen. These results have implications for the computational complexity of diagonal extraction for D-finite series and connect to broader themes in algebraic and differential-algebraic approaches to diagonals across characteristic zero.
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.