Algebraically constructible functions: real algebra and topology

TL;DR

This survey explores algebraically constructible functions, highlighting their role in linking real algebra with topology, and discusses historical context, properties, and obstructions to topological realization as real algebraic sets.

Contribution

It provides a comprehensive overview of algebraically constructible functions, including their definitions, properties, algebraic characterizations, and topological obstructions.

Findings

Connections between real algebra and topology are elucidated.

Key properties and algebraic characterizations are summarized.

Obstructions to topological realization as real algebraic sets are described.

Abstract

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set.