Computing transcendence and linear relations of 1-periods

TL;DR

This paper presents an algorithm to compute all linear relations among 1-periods, decide their transcendence, and determine equality, thereby solving a problem posed by Kontsevich and Zagier for periods.

Contribution

We develop a practical algorithm for analyzing 1-periods, including their linear relations, transcendence, and equality, based on the theory of 1-motives and divisor arithmetic.

Findings

Algorithm determines all linear relations among 1-periods

Decides whether a 1-period is transcendental

Classifies autonomous first-order differential equations

Abstract

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite collection of 1-periods, computes the space of all linear relations among them with algebraic coefficients. In particular, the algorithm decides whether a given 1-period is transcendental, and whether two 1-periods are equal. This resolves, in the case of 1-periods, a problem posed by Kontsevich and Zagier, asking for an algorithm to decide equality of periods. The algorithm builds on the work of Huber and W\"ustholz, who showed that all linear relations among 1-periods arise from 1-motives; we make this perspective effective by reducing the problem to divisor arithmetic on curves and providing the theoretical foundations for a practical and fully explicit algorithm. To illustrate the broader applicability of our methods, we also give an algorithmic classification of autonomous first-order (non-linear) differential equations.