We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

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We establish a formula for the volume Poincaré series of a log smooth scheme. This yields in particular a new expression and a smaller set of candidate poles for the motivic zeta function of a hypersurface singularity and of a degeneration of Calabi-Yau varieties. RésuméCalcul de fonctions zêta à partir de modèles log lisses. Nous établissons une formule pour la série volume de Poincaré d'un schéma log lisse. Ceci nous fournit en particulier une nouvelle expression et un ensemble réduit de candidats pôles pour la fonction zêta motivique d'une singularité d'hypersurface et d'une dégénération de variétés de Calabi-Yau.