2018/05/18 by Bertrand Eynard, B. Eynard, Eynard, B. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Algebra over a field #Algebraic expression #Algebraic number #Computer science #Differential (mechanical device) #Differential algebraic equation #Differential algebraic geometry #Differential equation #Expression (computer science) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Ordinary differential equation #Physics #Polynomial and algebraic computation #Pure mathematics #Thermodynamics #hep-th #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1805.07247
18 pages, Latex. Some misprints corrected
openalex publication_date 2018/05/18 · arxiv created 2018/08/28 · arxiv updated 2018/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.