A proof of the Brill-Noether method from scratch

TL;DR

This paper provides an elementary, self-contained proof of the classical Brill-Noether method for computing Riemann-Roch space bases, avoiding complex algebraic geometry concepts.

Contribution

It introduces a new, simplified proof of the Brill-Noether method using elementary techniques like Newton polygons and Hensel lifting.

Findings

The proof is self-contained and elementary.

It relies on Newton polygons, Hensel lifting, and resultants.

The approach simplifies understanding of the classical method.

Abstract

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.