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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.

##### MSC:
 32S10 Invariants of analytic local rings
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##### References:
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