E\~ne product in the transalgebraic class

TL;DR

This paper introduces transalgebraic functions on Riemann surfaces, extending the e e product to this class, and explores their algebraic and divisor properties, linking to polylogarithm hierarchies.

Contribution

It defines the transalgebraic class, extends the e e product to it, and analyzes its algebraic structure and divisor actions, connecting to polylogarithm hierarchies.

Findings

Transalgebraic functions form a topological multiplicative group.

The e e product extends to the transalgebraic class on the Riemann sphere.

The class forms a commutative ring under addition and e e product.

Abstract

We define transalgebraic functions on a compact Riemann surface as meromorphic functions except at a finite number of punctures where they have finite order exponential singularities. This transalgebraic class is a topological multiplicative group. We extend the action of the e\~ne product to the transcendental class on the Riemann sphere. This transalgebraic class, modulo constant functions, is a commutative ring for the multiplication, as the additive structure, and the e\~ne product, as the multiplicative structure. In particular, the divisor action of the e\~ne product by multiplicative convolution extends to these transalgebraic divisors. The polylogarithm hierarchy appears related to transalgebraic e\~ne poles of higher order.