Volume 18, issue 2 (2014)

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Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers

Georges Comte and Goulwen Fichou

Geometry & Topology 18 (2014) 963–996
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 this ring contains as a subring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows us to express a class as a $ℤ\left[\frac{1}{2}\right]$–linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincaré 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 fibers.

Keywords
Grothendieck ring, semialgebraic sets, motivic Milnor fiber
Mathematical Subject Classification 2010
Primary: 14P10
Secondary: 14B05, 14P25
Publication
Received: 4 September 2012
Revised: 10 October 2013
Accepted: 14 November 2013
Published: 7 April 2014
Proposed: Lothar Goettsche
Seconded: Richard Thomas, Jim Bryan
Authors
 Georges Comte Laboratoire de Mathématiques Université de Savoie UMR CNRS 5127 Bâtiment Chablais, Campus scientifique 73376 Le Bourget-du-Lac France http://gc83.perso.sfr.fr/ Goulwen Fichou IRMAR Université de Rennes 1 UMR CNRS 6625 Campus de Beaulieu 35042 Rennes France http://perso.univ-rennes1.fr/goulwen.fichou/