Petrov modules and zeros of Abelian integrals

✍ Scribed by Lubomir Gavrilov

• Book ID: 104106368
• Publisher: Elsevier Science
• Year: 1998
• Tongue: French
• Weight: 621 KB
• Volume: 122
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/s0007-4497(99)80004-9

✦ Synopsis

We prove that the Petrov module Pf associated to an arbitrary semiweighted homogeneous polynomial f E C[X, y] is free and finitely generated. We compute its generators and use this to obtain a lower bound for the maximal number of zeros of complete Abelian integrals. 0 Elsevier, Paris

1. Statement of the results

Let f E C[x:, y] be a polynomial and consider the quotient vector space Pf of polynomial one-forms w = P&r+ Qdy, modulo one-forms dA+ Bdf where A, B are polynomials. Pi is a module over the ring of polynomials C[t], under the multiplication R(t) . w = R(f)w.

Recall that a function f : C2-+ C is called weighted homogeneous (wh) of weighted degree d and type w = (w,~, wy), w, = weight(z), WY = weight(y) if

(1) f(Pk, .iY'y y) = zdf(z, y), v.2 E c*.