On arithmetic of plane trees

TL;DR

This paper extends the study of the arithmetic properties of plane bipartite trees, establishing a lower bound on the degree of their definition field under specific divisibility conditions related to a prime number.

Contribution

It generalizes previous results by providing a new lower bound for the degree of the definition field for trees with certain prime divisibility properties.

Findings

Established a lower bound on the degree of the definition field for specific plane bipartite trees.

Identified conditions involving prime divisibility related to the structure of the trees.

Extended the arithmetic analysis of plane trees beyond previous prime cases.

Abstract

In [3] L.Zapponi studied the arithmetic of plane bipartite trees with prime number of edges. He obtained a lower bound on the degree of tree's definition field. Here we obtain a similar lower bound in the following case. There exists a prime number $p$ such, that: a) the number of edges is divisible by $p$, but not by $p^2$; b) for any proper subset of the set of white (or black) vertices the sum of their degrees is not divisible by this $p$.