A proof of the Brill-Noether method from scratch

TL;DR

The paper advances Brill and Noether’s classical method for computing Riemann--Roch spaces by delivering a short, self-contained proof that relies on Newton polygons, Hensel lifting, and resultant techniques. It builds a complete framework from valuations, places, and divisors on plane curves to a residue-based Brill--Noether algorithm that produces a common denominator $H$ for $\mathcal{L}(D)$ and a basis of homogeneous numerators, $G_i$, modulo the curve equation. Central to the approach is the adjoint divisor $\mathcal{A}$, which guides the divisor inequalities used to guarantee the existence of $H$ and the correct numerator basis; the residue theorem then bridges local and global conditions. The work also discusses algorithmic aspects, implementation considerations, and connections to algebraic-geometry coding theory (AG codes) and secret sharing, highlighting practical applications and extensions to general curves in various characteristics.

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.

Figures (1)

• Figure 1: Newton polygon of $f = x^3 y + 2 xy^2 - x^2 y^4 + y^5 + 3 xy^6 + y^7 \in \mathbb{Q} [[x]] [y]$.

Theorems & Definitions (34)

• Remark 1
• Example 1
• Remark 2
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 1
• Lemma 2
• Proposition 5