Combinatorial expression of the fundamental second kind differential on an algebraic curve
Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 2, pp. 219-238.
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The zero locus of a bivariate polynomial P(x,y)=0 defines a compact Riemann surface Σ. The fundamental second kind differential is a symmetric 1⊗1-form on Σ×Σ 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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/116
Classification : 14-XX
Mots-clés : algebraic geometry, Riemann surfaces, Newton polygon 
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     author = {Eynard, Bertrand},
     title = {Combinatorial expression of the fundamental second kind differential on an algebraic curve},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {219--238},
     volume = {9},
     number = {2},
     year = {2022},
     doi = {10.4171/aihpd/116},
     mrnumber = {4450014},
     zbl = {1492.14054},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/aihpd/116/}
}
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Eynard, Bertrand. Combinatorial expression of the fundamental second kind differential on an algebraic curve. Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 2, pp. 219-238. doi : 10.4171/aihpd/116. http://www.numdam.org/articles/10.4171/aihpd/116/

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