Combinatorial expression of the fundamental second kind differential on an algebraic curve
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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 .
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|---|---|---|
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