A generalization of Baker's theorem

Baker's theorem is a theorem giving an upper-bound for the genus of a plane curve. It can be obtained by studying the Newton-polygon of the defining equation of the curve. In this paper we give a different proof of Baker's theorem not using Newton-polygon theory, but using elementary methods from the theory of function fields (Theorem 2.4). Also we state a generalization to several variables that can be used if a curve is defined by several bivariate polynomials that all have one variable in common (Theorem 3.3). As a side result, we obtain a partial explicit description of certain Riemann–Roch spaces, which is useful for applications in coding theory. We give several examples and compare the bound on the genus we obtain, with the bound obtained from Castelnuovo's inequality.