Bounding the Degree of Belyi Polynomials

TL;DR

This paper investigates the minimal degree of Belyi polynomials mapping algebraic numbers to zero or one, establishing lower bounds based on p-adic valuations and providing examples to demonstrate the bounds' sharpness.

Contribution

It introduces the concept of Belyi height for algebraic numbers and proves lower bounds for this height using Newton polygon combinatorics.

Findings

Belyi height is at least p for algebraic numbers with non-zero p-adic valuation.

Examples show the bounds are sharp.

Provides a combinatorial approach to bounding Belyi polynomial degrees.

Abstract

Belyi's Theorem states that a Riemann surface, X, as an algebraic curve is defined over an algebraic closure of the rationals if and only if there exists a holomorphic function taking X to the Riemann sphere with at most three critical values (traditionally taken to be zero, one, and infinity). By restricting to the case where X is the Riemann sphere and our holomorphic functions are Belyi polynomials, we define a Belyi height of an algebraic number to be the minimal degree of Belyi polynomials mapping said algebraic number to either zero or one. We prove, for non-zero algebraic numbers with non-zero p-adic valuation, that the Belyi height must be greater than or equal to p using the combinatorics of Newton polygons. We also give examples of algebraic numbers which show our bounds are sharp.