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

The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $\mathrm{\Sigma }$. The fundamental second kind differential is a symmetric $1\otimes 1$ form on $\mathrm{\Sigma }\times \mathrm{\Sigma }$ 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 $P$.