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
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
