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 $(\mathbb C^d,f^{-1}(0))$.When $d=2,3$ this result holds by assuming merely the topological type insteadof the analytic one. G. Fichou and T. Fukui recently proved the following real counterpart: theblow-Nash type of a real singular non-degenerate convenient weightedhomogeneous polynomial in three variables determines its weights. The aim of this paper is to generalize the above-cited result with nocondition on the number of variables. We work with a characterization of theblow-Nash equivalence called the arc-analytic equivalence. It is an equivalencerelation on Nash function germs with no continuous moduli which may be seen asa semialgebraic version of the blow-analytic equivalence of T.-C. Kuo.