Bounding the degree of Belyi polynomials

TextBelyiʼs theorem states that a Riemann surface X  , as an algebraic curve, is defined over Q¯ if and only if there exists a holomorphic function B taking X   to P1CP1C with at most three critical values {0,1,∞}{0,1,∞}. By restricting to the case where X=P1CX=P1C and our holomorphic functions are Belyi polynomials, for an algebraic number λ  , we define a Belyi height H(λ)H(λ) to be the minimal degree of the set of Belyi polynomials with B(λ)∈{0,1}B(λ)∈{0,1}. We prove for non-zero λ with non-zero p-adic valuation, the Belyi height of λ is greater than or equal to p using the combinatorics of Newton polygons. We also give examples of algebraic numbers with relatively low height and show that our bounds are sharp.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=MJAodACJ4kM.