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arXiv · 1805.07247

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

B. Eynard
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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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B. Eynard. 2018-05-18. Notes about a combinatorial expression of the fundamental second kind differential on an algebraic curve. https://arxiv.org/abs/1805.07247

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