The algebra of balanced dessins

TL;DR

This paper introduces the algebra of balanced dessins, a new framework aiming to systematically associate algebraic numbers with dessins through formal sums, bridging combinatorics and algebraic number theory.

Contribution

It defines the algebra of balanced dessins and proposes a novel approach to connect dessins with algebraic numbers systematically.

Findings

Defined the algebra of balanced dessins using formal sums.

Established a framework for associating algebraic numbers with dessins.

Provided foundational concepts for future systematic constructions.

Abstract

This paper gives a key definition, for a new approach to dessins and algebraic numbers. The distant goal is to construct from each dessin $D$ an algebraic number $\eta_D$, in a systematic and useful way. The algebra of balanced dessins is generated by formal sums $\psi_D$ of dessins, intended to be intermediate between $D$ and $\eta_D$.