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We define a Grothendieck ring for basic real semialgebraic formulas, that isfor systems of real algebraic equations and inequalities. In this ring theclass of a formula takes into consideration the algebraic nature of the set ofpoints satisfying this formula and contains as a ring the usual Grothendieckring of real algebraic formulas. We give a realization of our ring that allowsto express a class as a Z[1/2]- linear combination of classes of real algebraicformulas, so this realization gives rise to a notion of virtual Poincar\'epolynomial for basic semialgebraic formulas. We then define zeta functions withcoefficients in our ring, built on semialgebraic formulas in arc spaces. Weshow that they are rational and relate them to the topology of real Milnorfibres.

Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres