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The infinitesimal 16th Hilbert problem in dimension zero. (English) Zbl 1119.14040

Let \(X\) and \(Y\) be Zariski open subsets in \({\mathbb C}^{q+1}\) and \({\mathbb C}\) respectively, \(f : X \rightarrow Y\) a morphism given by a polynomial, \(\gamma(a) \in H_q(f^{-1}(a), {\mathbb Z})\) a continuous family of \(q\)-cycles and \(\omega\) a polynomial \(q\)-form in \({\mathbb C}^{q+1}\), all these objects being defined over a subfield \(k \subset {\mathbb C}\). The infinitesimal 16th Hilbert problem asks to find the exact upper bound \(Z(m,n,k,q)\) for the number of zeros \(a \in K \cap D\) of the abelian integral \(I(a)= \int_{\gamma(a)} \omega\) where \(\deg(f) \leq m\), \(\deg(\omega) \leq n\) and \(D\) is any simply connected domain in \(Y\).
In this paper, the authors consider the case \(q=0\) and its main result is Theorem 1 where they prove \[ m-1 - \lfloor \tfrac{n}{m} \rfloor \leq Z(m,n,k,0) \leq \frac{(m-1)(n-1)}{2}. \]
Moreover, it is also investigated some arithmetic properties of abelian integrals. Thus, they improve the upper bound for the number of zeros of an abelian integral with fixed \(f\) and give a necessary and sufficient condition for an abelian integral to be zero.

MSC:

14K20 Analytic theory of abelian varieties; abelian integrals and differentials
32S65 Singularities of holomorphic vector fields and foliations
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)

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References:

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