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

Understand

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 $P$.