On the arc-analytic type of some weighted homogeneous polynomials

摘要

It is known that the weights of a complex weighted homogeneous polynomial f with isolated singularity are analytic invariants of (Cd,f-1(0)). When d=2,3 this result holds by assuming merely the topological type instead of the analytic one. Fichou and Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights. The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a characterization of the blow-Nash equivalence called the arc-analytic equivalence. It is an equivalence relation on Nash function germs with no continuous moduli which may be seen as a semialgebraic version of the blow-analytic equivalence of Kuo.