Bounding Belyi Polynomials and their relation to Dessin D'enfant A Belyi function is a function from an algebraic curve to the Riemann sphere that is ramified over 3 points. They are important in Grothendieck's dessin d'enfant program. We study the case when Belyi functions are also polynomials. By employing degeneration techniques, we find that the ramification condition on Belyi functions puts strong constraints on the Newton polygons of these polynomials. Using this we can find lower bounds on the degree of a Belyi polynomial that maps certain rational numbers to particular points on the Riemann sphere.