Multidimensional generalization of the Il'yashenko theorem on abelian integrals (Q1392329)

The main theorem proved in this paper is as follows: Let \(P(x_{1},\dots,x_k\) be a real polynomial of degree \(n>k\) with all complex critical points being Morse and all critical values distinct. Let \(P=0\) be a noncritical level surface and \(\alpha\) a polynomial \((k-1)\)-form with coefficients of degree \(\leq n-k+1\). Let \(\gamma(0)\) be a compact connected component of the real surface \(P=0\) and \(\gamma(c) \subset (P=0)\) its continuous deformation. If the identity \(\int_{\gamma(c)} \alpha \equiv 0\) holds for all sufficiently small values of parameter \(c\), then \(\alpha\) is exact. This result generalizes a theorem of Il'yashenko on the zeros of Abelian integrals to higher dimensions. The approach adopted by the author is based on singularity theory and algebraic geometry which differs from Il'yashenko's.