[Paper Review] Minimization of differential equations and algebraic values of $E$-functions

Alin Bostan, Tanguy Rivoal+1 more|arXiv (Cornell University)|Sep 5, 2022

Mathematical Approximation and Integration4 citations

TL;DR

This paper presents efficient algorithms for minimizing linear differential equations—finding the lowest-order homogeneous or inhomogeneous differential operator with rational coefficients that annihilates a given power series. By combining modular arithmetic, approximant basis computation, and certified reconstruction, the method enables the determination of algebraic values of $E$-functions, proving that certain Apéry-like $E$-functions are purely transcendental and supporting conjectures on their minimal order and degree structures.

ABSTRACT

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.

Motivation & Objective

• To develop efficient algorithms for minimizing linear differential equations with rational coefficients, both in homogeneous and inhomogeneous forms, for power series solutions.
• To apply these minimization techniques to solve problems in transcendental number theory, particularly identifying algebraic points where $E$-functions take algebraic values.
• To provide a certified, computationally efficient framework for determining the minimal-order differential operator of an $E$-function, leveraging modular arithmetic and rational reconstruction.
• To extend and verify conjectures on the structure of minimal differential operators for Apéry-like $E$-functions, including order and degree formulas.
• To implement and benchmark the algorithms on families of $E$-functions defined by hypergeometric-type series, demonstrating scalability and reliability.

Proposed method

• The method uses modular reduction of the power series coefficients modulo a prime to compute approximant bases, enabling efficient reconstruction of the minimal differential operator.
• It applies rational reconstruction techniques to recover the exact polynomial coefficients of the minimal differential operator from modular images.
• The algorithm combines tools from van Hoeij's factorization framework with additional constraints derived from the known power series solution to accelerate minimization.
• Certification of minimality is achieved by verifying that no lower-order operator exists via exhaustive search in the modular setting and reconstruction.
• The approach handles both ordinary and singular points, including apparent singularities, by carefully managing initial conditions and series expansions.
• The implementation integrates Zeilberger's algorithm for recurrence computation and Beukers' matrix method to support the minimization pipeline.

Experimental results

Research questions

• RQ1 Is the $E$-function $f_{m,p}$ purely transcendental for all $m \geq 1$ and $p \geq 1$?
• RQ2 Does the minimal-order differential equation for $f_{m,p}$ have order $\lfloor(N+1)^2/4\rfloor$ and degree $\lfloor N(2N^2 - 3N + 4)/12 \rfloor$, where $N = m + p$?
• RQ3 When both $m$ and $p$ are odd, is $\operatorname{ord}(L^\textrm{min}_{m,p}) = \operatorname{ord}(L^\textrm{min}_{p,m})$ and $\deg(L^\textrm{min}_{m,p}) = \deg(L^\textrm{min}_{p,m})$?
• RQ4 Can the minimal differential operator for $f_{m,p}$ be computed efficiently and certified as minimal using modular and rational reconstruction techniques?
• RQ5 What proportion of the total computation time is spent on certifying minimality, and how does this vary with the complexity of the series?

Key findings

• For the family $f_{m,p}(z) = \sum_{n=0}^\infty \left( \sum_{k=0}^n \binom{n}{k}^m \binom{n+k}{k}^p \right) \frac{z^n}{n!}$, the minimal homogeneous differential equation was successfully computed for $(m,p)$ up to $(5,5)$, with timings ranging from 2.5 seconds to over 14 minutes.
• The proportion of time spent on certifying minimality ranged between 20% and 30% of the total runtime, indicating that this step is computationally significant but manageable.
• The most time-consuming phase was the computation of the approximant basis over rational coefficients, accounting for up to half of the total time, especially in high-degree cases.
• Modular computation of approximant bases was faster than rational reconstruction, consuming between 1/5 and 1/3 of the total time, despite handling larger series orders.
• The rational coefficient reconstruction step became the most expensive operation in the largest example $(m,p)=(5,5)$, requiring over 4,500 rational coefficients to be computed.
• The results support the conjecture that $f_{m,p}$ is purely transcendental for all $m,p \geq 1$, extending previous results for $(m,p)=(2,1)$.