A Proof of the Brill-Noether Method from Scratch

In 1874 Brill and Noether designed a seminal geometric method for computing bases of Riemann-Roch spaces. From then, their method has led to several algorithms, some of them being implemented in computer algebra systems. The usual proofs often rely on abstract concepts of algebraic geometry and commutative algebra. In this paper we present a short self-contained and elementary proof that mostly needs Newton polygons, Hensel lifting, bivariate resultants, and Chinese remaindering.