2013/12/23 by Morihiko Saito, Saito, Morihiko · 2 citations
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Argument (complex analysis) #Convergence (economics) #Discrete mathematics #FOS: Mathematics #Mathematical analysis #Mathematics #Polynomial #Polynomial and algebraic computation #Pure mathematics #Simple (philosophy) #Uniqueness #math.AG
paper · pdf · doi:10.48550/arxiv.1312.6629
23 pages
openalex publication_date 2013/12/23 · arxiv created 2018/01/20 · arxiv updated 2018/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a simple proof of the uniqueness of extensions of good sections for formal Brieskorn lattices, which can be used in a paper of C. Li, S. Li, and K. Saito for the proof of convergence in the non-quasihomogeneous polynomial case. Our proof uses an exponential operator argument as in their paper, although we do not use polyvector fields nor smooth differential forms. We also present an apparently simpler algorithm for an inductive calculation of the coefficients of primitive forms in the Brieskorn-Pham polynomial case. In a previous paper on the structure of Brieskorn lattices, there were some points which were not yet very clear, and we give some explanations about these, e.g. on the existence and the uniqueness of primitive forms associated with good sections, where we present some rather interesting examples. In Appendix we prove the uniqueness up to a nonzero constant multiple of the higher residue pairings in some formal setting which is different from the one in the main theorem. This is questioned by D. Shklyarov.