The diagonal and Hadamard grade of hypergeometric functions
Andrew Harder, Joe Kramer-Miller
TL;DR
The paper introduces the diagonal grade $\textbf{dg}(f)$ and the Hadamard grade $\textbf{hg}(f)$ as measures of how far a power series is from being a diagonal or Hadamard product of diagonals, and it connects these notions to the nilpotence of the associated differential equations via Deligne’s open monodromy theorem and Picard–Fuchs theory. It proves universal bounds linking $\textbf{dg}(f)$ and $\textbf{hg}(f)$ to the local monodromy, and applies these bounds to a broad class of hypergeometric functions, obtaining exact grades in many cases, including the crucial result that $\prescript{}{n}F_{n-1}(1/2,\dots, 1/2; 1,\dots, 1 \mid x)$ has diagonal grade $n$ and that the Apéry generating function has diagonal grade $3$. The work establishes strict inclusions $\mathcal{D}_k \subsetneq \mathcal{D}_{k+1}$ and $\mathcal{H}_k \subsetneq \mathcal{H}_{k+1}$ and provides a detailed description of the ring of diagonals under Hadamard multiplication, including when zero divisors occur via the $\mu_m$-action. These results advance the understanding of diagonals beyond the previously known cases, contribute unconditional bounds to questions about Hadamard grades, and connect to broader themes in algebraic geometry, Calabi–Yau geometry, and quantum field theory.
Abstract
Diagonals of rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. In this paper we study the diagonal grade of a function $f$, which is defined to be the smallest $n$ such that $f$ is the diagonal of a rational function in variables $x_0,\dots, x_n$. We relate the diagonal grade of a function to the nilpotence of the associated differential equation. This allows us to determine the diagonal grade of many hypergeometric functions and answer affirmatively the outstanding question on the existence of functions with diagonal grade greater than $2$. In particular, we show that $\prescript{}{n}F_{n-1}(\frac{1}{2},\dots, \frac{1}{2};1\dots,1 \mid x)$ has diagonal grade $n$ for each $n\geq 1$. Our method also applies to the generating function of the Apéry sequence, which we find to have diagonal grade $3$. We also answer related questions on Hadamard grades posed by Allouche and Mendès France. For example, we show that $\prescript{}{n}F_{n-1}(\frac{1}{2},\dots, \frac{1}{2};1\dots,1 \mid x)$ has Hadamard grade $n$ for all $n\geq 1$.