The Ramificant Determinant

Kingshook Biswas (1) , Ricardo Pérez-Marco (2)

1 Indian Statistical Institute [Kolkata]
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus 0). We define the base vector space of transcendental functions and establish by elementary means some transcen-dental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points.

Domaines: Variables complexes [math.CV]; Théorie des nombres [math.NT]

Kingshook Biswas, Ricardo Pérez-Marco. The Ramificant Determinant. Symmetry, Integrability and Geometry : Methods and Applications, 2019. ⟨hal-02072156v2⟩