## Finiteness on semialgebraic types of Nash mappings defined on a Nash surface

Satoshi Koike (Hyogo)

Abstract

We consider a problem on whether the number of semialgebraic types appearing in a family of Nash mappings defined on a 2-dimensional Nash surface is finite. We show that if the Nash surface has isolated singularities, then the number is finite. On the other hand, we show that if it has non-isolated singularities, namely its singular locus is one-dimensional, then the number can be infinite. We give such a negative example in the algebraic case. This is a joint work with Masahiro Shiota.