Finiteness theorem on Blow-semialgebraic triviality for a family of 3-dimensional algebraic sets

TL;DR

This paper establishes a finiteness theorem for Blow-semialgebraic triviality in families of 3-dimensional algebraic and Nash sets, providing a form of equisingularity and classification in real algebraic geometry.

Contribution

It introduces Blow-semialgebraic triviality with compatible filtration and proves finiteness results for families of 3D algebraic and Nash sets, extending to polynomial mappings.

Findings

Finite subdivision into Nash manifolds for trivialization

Finiteness of semialgebraic types of polynomial mappings

Extension of results to Nash families via Artin-Mazur theorem

Abstract

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 3-dimensional algebraic sets defined over a nonsingular 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 trivialisation consistent with a compatible filtration. We show a similar result on finiteness also for a Nash family of 3-dimensional Nash sets through the Artin-Mazur theorem. As a corollary of the arguments in their proofs, we have a finiteness theorem on semialgebraic types of polynomial mappings from the 2-dimensional Euclidean space to the p-diemnsional Euclidean space.