Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers

Comte, Georges¹ ; Fichou, Goulwen²

¹ Laboratoire de Mathématiques, Université de Savoie, UMR CNRS 5127, Bâtiment Chablais, Campus scientifique, 73376 Le Bourget-du-Lac, France
² IRMAR, Université de Rennes 1, UMR CNRS 6625, Campus de Beaulieu, 35042 Rennes, France

Geometry & topology, Tome 18 (2014) no. 2, pp. 963-996.

Résumé

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 $\mathbb{Z}\left[\frac{1}{2}\right]$–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.

Détail

DOI : 10.2140/gt.2014.18.963

Classification : 14P10, 14B05, 14P25
Keywords: Grothendieck ring, semialgebraic sets, motivic Milnor fiber

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