Equivariant zeta functions for invariant Nash germs

Fabien Priziac (1, 2)

1 I2M - Institut de Mathématiques de Marseille
2 Department of Mathematics, Faculty of Science, Saitama University

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef-Loeser formulae for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

Mots clés: Nash germs; group action; blow-Nash equivalence; zeta functions; equivariant virtual Poincaré series; Denef-Loeser formula

Domaines: Géométrie algébrique [math.AG]

Fabien Priziac. Equivariant zeta functions for invariant Nash germs. Nagoya Mathematical Journal, 2016, 222 (01), pp 100 - 136. ⟨10.1017/nmj.2016.12⟩. ⟨hal-00955670v2⟩