Arc-analytic functions on locally closed semialgebraic sets

Abstract

We give a generalization of the Kurdyka theory of arc-analytic functions and arc-symmetric sets on Nash manifolds to the setting of arbitrary locally closed semialgebraic sets. In particular, we prove that the ideals of the ring of arc-analytic functions on a locally closed set $S$ satisfy an arc-analytic Nullstellensatz, their zero-sets are precisely the semialgebraic arc-symmetric subsets of $S$, which form the family of closed sets of a Noetherian topology on $S$, and the Krull dimension of the ring is equal to the Euclidean dimension of $S$ and to the Krull dimension of $S$ in that topology.