Higher order approximation of analytic sets by topologically equivalent algebraic sets

TL;DR

This paper demonstrates that analytic set germs can be approximated by algebraic germs with arbitrary tangency order, ensuring topological equivalence and close arc space approximation up to a specified truncation.

Contribution

It introduces a method to approximate analytic set germs by algebraic germs with any desired order of tangency, refining previous topological equivalence results.

Findings

Homeomorphism with prescribed tangency order between analytic and algebraic germs

Approximation of arc spaces up to a specified truncation order

Enhanced understanding of the relationship between analytic and algebraic sets

Abstract

It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.