Notes about a combinatorial expression of the fundamental second kind differential on an algebraic curve

TL;DR

This paper presents a combinatorial formula for the fundamental second kind differential on an algebraic curve, expressed solely through the Newton polygon's combinatorics and polynomial coefficients.

Contribution

It introduces a rational, combinatorial expression for the differential using only integer combinations of polynomial coefficients, avoiding complex analysis.

Findings

Provides a rational formula based on Newton's polygon

Uses only combinatorics and polynomial coefficients

Applicable over the same field as the polynomial coefficients

Abstract

The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $\Sigma$. The fundamental second kind differential is a symmetric $1\otimes 1$ form on $\Sigma\times \Sigma$ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of $P$, involving only integer combinations of products of coefficients of $P$. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of $P$.