Abstract
Belyiʼs 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 P 1 C with at most three critical values { 0 , 1 , ∞ } . By restricting to the case where X = P 1 C and our holomorphic functions are Belyi polynomials, for an algebraic number λ , we define a Belyi height H ( λ ) to be the minimal degree of the set of Belyi polynomials with 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. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=MJAodACJ4kM . Author Video Watch what authors say about their articles
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