Publication : t18/052

Abstract:
The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $curve$. The fundamental second kind differential is a symmetric $1otimes 1$ form on $curvetimes curve$ 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 $P$.

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