Movasati, Hossein Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations. (English) Zbl 1117.14014 Brasselet, Jean-Paul (ed.) et al., Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19–23, 2004. Basel: Birkhäuser (ISBN 3-7643-7775-5/hbk). Trends in Mathematics, 247-262 (2007). The paper introduces algorithms for computing iterations of Gauss-Manin connections, Picard-Fuchs equations of abelian integrals and mixed Hodge structures of affine varieties of dimension \(n\) in terms of differential forms. For \(n=1\) such computations can be applied to counting the limit cycles of planar systems of polynomial differential equations. For \(n>3\) they give explicit definitions of Hodge cycles.For the entire collection see [Zbl 1108.14001]. Reviewer: Vladimir P. Kostov (Nice) Cited in 6 Documents MSC: 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32S65 Singularities of holomorphic vector fields and foliations 14D07 Variation of Hodge structures (algebro-geometric aspects) 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) Keywords:Picard-Fuchs equation; abelian integral; Gauss-Manin connection; mixed Hodge structure; Hodge cycle Software:foliation.lib PDFBibTeX XMLCite \textit{H. Movasati}, in: Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19--23, 2004. Basel: Birkhäuser. 247--262 (2007; Zbl 1117.14014) Full Text: arXiv