2024/10/28 by Evelia R. García Barroso, Barroso, Evelia Rosa García, Patrick Popescu‐Pampu +1
Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2410.21250
openalex publication_date 2024/10/28 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28
Let (f,g): (S,s) → (ℂ2, 0) be a finite morphism from a germ of normal complex analytic surface to the germ of ℂ2 at the origin. We show that the affine algebraic curve in ℂ2 defined by the initial Newton polynomial of a defining series of the discriminant germ of (f,g) depends up to toric automorphisms only on the germs of curves defined by f and g. This result generalizes a theorem of Gryszka, Gwoździewicz and Parusiński, which is the special case in which (S,s) is smooth. Our proof uses a common generalization of formulas of Lê, Casas-Alvero and Némethi for the intersection number of the discriminant with a germ of plane curve. It uses also a theorem of Delgado and Maugendre characterizing the special members of pencils of curves on normal surface singularities. We apply it to the pencils generated by all pairs (fb, ga), for varying positive integral exponents a, b, following a strategy initiated by Gwoździewicz and by Delgado and Maugendre. This is similar to the Eudoxian method of comparison of magnitudes by comparing the sizes of their positive integral multiples.