Logarithmic limit sets of real semi-algebraic sets

TL;DR

This paper investigates the properties of logarithmic limit sets of real semi-algebraic and o-minimal definable sets, establishing parallels with complex algebraic sets and connecting to tropical geometry and non-archimedean fields.

Contribution

It extends key properties of logarithmic limit sets from complex algebraic sets to real semi-algebraic and o-minimal structures, highlighting their polyhedral nature and geometric relations.

Findings

Logarithmic limit sets of real semi-algebraic sets are polyhedral.

Properties of complex algebraic sets' limit sets extend to the real case.

Connections to tropical geometry and non-archimedean fields are established.

Abstract

This paper is about the logarithmic limit sets of real semi-algebraic sets, and, more generally, about the logarithmic limit sets of sets definable in an o-minimal, polynomially bounded structure. We prove that most of the properties of the logarithmic limit sets of complex algebraic sets hold in the real case. This include the polyhedral structure and the relation with the theory of non-archimedean fields, tropical geometry and Maslov dequantization.