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Register a new userNotes about a combinatorial expression of the fundamental second kind differential on an algebraic curve
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Authors
B. Eynard
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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 $1otimes 1$ form on $Sigmatimes Sigma$ 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 Newtons 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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