无穷小希尔伯特第16问题的限制版本
Restricted version of the infinitesimal Hilbert 16th problem
摘要：
本文讨论{ \it无穷小Hilbert第16问题}：求一个Abel积分零点个数的上估计，该积分被看作一个参数的函数。更详细地，考虑平面中次数为$ n+1 $的实多项式$ H$，以及该多项式的连续椭圆族$γ_t$（水平曲线$ H = t$的紧凑分量）。考虑多项式1-形式的$ω$，其次数系数至多为$n.$设I（t）= \int_{γ_t} ω. \label{I}问题是要给出这个积分的零点个数的上限估计。我们解决了这个问题的{ \it限制版本}。即，形式$ ω$是{ \it arbitrary，}，并且多项式$ H$虽然具有任意次数，但不太接近退化（非超莫尔斯）多项式的超曲面。我们希望问题的限制版本的解决方案是解决完整（非限制）版本的一步。
英文摘要：
The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial $ H$ of degree $ n+1 $ in the plane, and a continuous family of ovals $γ_t$ (compact components of level curves $ H = t$) of this polynomial. Consider a polynomial 1-form $ω$ with coefficients of degree at most $n.$ Let I(t) = \int_{γ_t} ω. \label{I} The problem is to give an upper estimate of the number of zeros of this integral. We solve a {\it restricted version} of this problem. Namely, the form $ ω$ is {\it arbitrary,}, and the polynomial $ H$, though having an arbitrary degree, is not too close to the hypersurface of degenerate (non ultra-Morse) polynomials. We hope that the solution of the restricted version of the problem is a step to the solution of the complete (nonrestricted) version.
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