Grothendieck Ring of Semialgebraic Formulas and Motivic Real Milnor Fibres

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 for instance 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 fibres.