On the arc-analytic type of some weighted homogeneous polynomials

TL;DR

This paper generalizes the understanding of how the arc-analytic type of weighted homogeneous polynomials determines their weights, extending previous results to any number of variables without restrictions.

Contribution

It introduces a broader characterization of arc-analytic equivalence to identify weights of weighted homogeneous polynomials in any number of variables.

Findings

Arc-analytic type determines weights of weighted homogeneous polynomials.

Generalization to any number of variables without restrictions.

Extends previous results from three variables to arbitrary dimensions.

Abstract

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 instead of the analytic one. G. Fichou and T. 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 T.-C. Kuo.