[Paper Review] Computing periods of rational integrals

Pierre Lairez|Apr 20, 2014

Advanced Differential Equations and Dynamical Systems4 citations

TL;DR

This paper presents a novel reduction algorithm extending the Griffiths-Dwork method to compute Picard-Fuchs equations for periods of rational integrals depending on a parameter. By iteratively reducing pole orders via a family of operations $[\,"]_r$, the method efficiently computes linear differential operators annihilating the period integral, enabling the solution of previously intractable problems with high efficiency and low memory usage on standard hardware.

ABSTRACT

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.

Motivation & Objective

• To develop a more efficient and practical algorithm for computing Picard-Fuchs equations of periods of rational integrals depending on a parameter.
• To overcome the limitations of general-purpose holonomic integration methods by exploiting the special structure of rational functions.
• To provide an elementary, algorithmic framework that computes the minimal differential operator annihilating the period integral, or a left multiple of it.
• To demonstrate the method's effectiveness on previously infeasible examples, including high-degree and high-order Picard-Fuchs equations.
• To challenge assumptions about maximally unipotent monodromy and minimal order in the context of periods from reflexive polytopes.

Proposed method

• Introduces a family of reductions $[\,"]_r$ for rational functions that differ from the original only by total derivatives, with $[\,"]_1$ being the classical Griffiths-Dwork reduction.
• Applies the reduction process iteratively to reduce the pole order of the rational function, ultimately expressing the integral as a total derivative if the period vanishes.
• Uses the condition $\mathcal{L}(R) = \sum \partial_i(B_i)$ to characterize the Picard-Fuchs operator $\mathcal{L}_R$, where $B_i$ are rational functions with bounded denominators.
• Employs symbolic computation to solve the linear system arising from the reduction process, leading to the minimal differential operator.
• Utilizes monomial substitutions (e.g., $x \mapsto 1/x$) to reduce the degree of the denominator, significantly improving computational performance.
• Employs van Hoeij's algorithmic techniques to test minimality of computed operators by bounding possible lower-order factors.

Experimental results

Research questions

• RQ1 Can a systematic reduction process be designed to compute Picard-Fuchs equations for rational integrals more efficiently than existing general holonomic methods?
• RQ2 Does the proposed reduction framework allow for the computation of previously infeasible Picard-Fuchs equations, especially those of high order or degree?
• RQ3 Are the minimal Picard-Fuchs operators for periods from reflexive polytopes always of order 4 with maximally unipotent monodromy, as commonly assumed?
• RQ4 Can degree-reducing monomial substitutions be systematically discovered or exploited to improve computational efficiency?
• RQ5 What is the minimal order of the Picard-Fuchs operator for periods arising from specific topologies of reflexive polytopes?

Key findings

• The algorithm computed a Picard-Fuchs equation of order 4 and degree 29 for a period from a 3-fold with 137 monomials in the denominator, completing in 80 seconds and using 30 MB of memory on a laptop.
• The method successfully computed a 4th-order Picard-Fuchs equation for the generating function of Apéry numbers, matching the known operator exactly.
• For a period from topology #27, the algorithm produced a 6th-order annihilating operator of degree 29, and it was proven that no lower-order minimal operator exists, contradicting expectations from Batyrev and Kreuzer.
• Among 137 newly computed periods, only one had a minimal Picard-Fuchs equation of order 4, indicating that such equations are rare and not guaranteed by polytope duality.
• The use of monomial substitutions reduced the denominator degree from 8 to 5, dramatically improving computation time, though optimal substitutions remain hard to predict.
• The method confirmed that the minimal operator for the period from topology #17 is of order 6, not 4, challenging the assumption that all such periods satisfy order-4 equations.