A 1-period is a complex number given by the integral of an algebraic function of a single variable, where both the function and the domain of integration are defined over algebraic numbers. We present 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, which asks for an algorithm to decide equality of periods. Our algorithm builds on the work of Huber and Wüstholz, 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 by developing 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.
2025. 54 pages.
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© 2025 Emre Can Sertöz, Joël Ouaknine, and James Worrell.