Multidimensional generalization of the Il'yashenko theorem on abelian integrals
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.
scientific article; zbMATH DE number 1179578