Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres

TL;DR

This paper introduces a Grothendieck ring for real semialgebraic formulas, enabling algebraic and topological analysis of real algebraic and semialgebraic sets, and relates zeta functions to Milnor fibers.

Contribution

It defines a new Grothendieck ring for basic real semialgebraic formulas and constructs zeta functions that connect algebraic formulas with topological invariants.

Findings

The ring captures algebraic and semialgebraic set classes.

Zeta functions are rational functions.

Connections to real Milnor fibers topology.

Abstract

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[1/2]- linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincar\'e polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibres.