Some classes of homeomorphisms that preserve multiplicity and tangent cones

TL;DR

This paper explores classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets, providing new insights into Zariski's questions and invariants in Lipschitz geometry.

Contribution

It introduces new classes of homeomorphisms preserving multiplicity and tangent cones, and offers effective approaches to Zariski's Question A, including at infinity.

Findings

Identified classes of homeomorphisms preserving multiplicity and tangent cones.

Provided new methods addressing Zariski's Question A.

Connected Lipschitz geometry with Nash modifications.

Abstract

In this paper we present some applications of A'Campo-L\^e's Theorem and we study some relations between Zariski's Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant when we consider right equivalence and this class contains many known classes of homeomorphisms that preserve tangent cones. In particular, we present some effective approaches to Zariski's Question A. We show a version of these results looking at infinity. Additionally, we present some results related with Nash modification and Lipschitz Geometry.