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Finiteness theorem for blow‐semialgebraic triviality of a family of three‐dimensional algebraic sets
Author(s) -
Koike Satoshi
Publication year - 2012
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds003
Subject(s) - mathematics , algebraic number , triviality , algebraic extension , corollary , dimension of an algebraic variety , real algebraic geometry , algebraic surface , function field of an algebraic variety , algebraic cycle , algebraic function , singular point of an algebraic variety , filtration (mathematics) , discrete mathematics , pure mathematics , mathematical analysis , differential algebraic equation , ordinary differential equation , differential equation
In this paper, we introduce the notion of ‘Blow‐semialgebraic triviality consistent with a compatible filtration’ for an algebraic family of algebraic sets, as an equisingularity for real algebraic singularities. Given an algebraic family of three‐dimensional algebraic sets defined over a non‐singular algebraic variety, we show that there is a finite subdivision of the parameter algebraic set into connected Nash manifolds over which the family admits a Blow‐semialgebraic trivialization consistent with a compatible filtration. We show a similar result for finiteness also of a Nash family of three‐dimensional Nash sets through the Artin–Mazur Theorem. As a corollary of the arguments in the proofs, we have a finiteness theorem for semialgebraic types of polynomial mappings from ℝ 2 to ℝ p .