Bounding the Degree of Belyi Polynomials

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.