Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 086, 28 pages      arXiv:1903.06770      https://doi.org/10.3842/SIGMA.2019.086
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

The Ramificant Determinant

Kingshook Biswas a and Ricardo Pérez-Marco b
a) Indian Statistical Institute, Kolkata, India
b) CNRS, IMJ-PRG, University Paris 7, Paris, France

Received March 13, 2019, in final form October 31, 2019; Published online November 05, 2019

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.

Key words: transalgebraic theory; Ramificant determinant; log-Riemann surface; Dedekind-Weber theory; ramified covering; exponential period; Liouville theorem.

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