# The Ramificant Determinant

## TL;DR

This paper develops a new theoretical framework for transalgebraic curves of genus zero, introducing the Ramificant Determinant and proving key theorems that enable algebraic reconstruction of these complex structures.

## Contribution

It introduces the Ramificant Determinant and establishes Abel-like and Torelli-like theorems for transalgebraic curves of genus zero, advancing the understanding of their transcendental properties.

## Key findings

- Defined the base vector space of transcendental functions.
- Derived a closed-form formula for the Ramificant Determinant.
- Proved Abel-like and Torelli-like theorems for transalgebraic curves.

## 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 means 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.