The Ramificant Determinant

Abstract

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 methods some transcendental 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. z 0 t e P 0 (t) dt, . . . , F d−1 (z) = z 0 t d−1 e P 0 (t) dt.

Eñe product in the transalgebraic class

“…More precisely, it is proved in [7] that any log-Riemann surface of finite topology (finitely generated fundamental group), is biholomorphic to a pointed compact Riemann surface X − S equipped with a transalgebraic differential form ω ∈ T Ω 1 (X), i.e. a differential form that is locally of the form ω = f (z)dz with f with exponential singularities, holomorphic out of S. Some of the transalgebraic properties of periods of transalgebraic curves are discussed in [8].…”