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Petrov modules and zeros of abelian integrals. (English) Zbl 0964.32022
The main results in this paper are the following theorems.
Theorem 1.1: Let \(f\in \mathbb{C}[x,y]\) be a semiweighted homogeneous polynomial of type \((w_x,w_y;d)\). The \(\mathbb{C}[t]\) module \({\mathcal P}_f\) (Petrov module) is free and finitely generated by \(\mu\) one-forms \(w_1, \dots,w_\mu\), where \(\mu =(d-w_x) (d-w_y)/w_xw_y\). Each one-form \(w_i\) can be defined by the condition \(dw_i =g_i dx\wedge dy\), where \(g_1,\dots,g_\mu\) is a monomial basis of the quotient ring \(\mathbb{C}[x,y]/f \langle f_x,f_y \rangle\). For every polynomial one-form \(w\) there exist polynomials \(a_k(t)\) of degree at most \((w\deg(w)-w\deg (w_k))/w\deg (f)\) such that in \({\mathcal P}_f\) there holds \(w=\sum^\mu_{k=1} a_k (t)w_k\).
Theorem 1.2: Let \(f\in\mathbb{C} [x,y]\) be a polynomial with only isolated critical points and suppose that for every \(t\in\mathbb{C}\) the fibre \(f^{-1} (t)\subset \mathbb{C}^2\) is connected. Every polynomial one-form \(w\) on \(\mathbb{C}^2\) satisfies the following condition: \[ \forall t\in\mathbb{C},\;w|_{f^{-1}(t)}= 0 \text{ in }H^1\bigl(f^{-1} (t)\bigr) \Leftrightarrow w=0 \text{ in }{\mathcal P}_f. \] Then the author uses these results to obtain a lower bound for the maximal number of zeros of complete abelian integrals. This problem was first stated by Arnold.

32S10 Invariants of analytic local rings
Full Text: DOI
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