Complete classification of Brieskorn polynomials up to the arc-analytic equivalence

TL;DR

This paper provides a complete classification of Brieskorn polynomials under arc-analytic equivalence, revealing how signs of coefficients influence their classification and extending previous results in the field.

Contribution

It introduces a new invariant for classifying Brieskorn polynomials up to arc-analytic equivalence, generalizing earlier classifications in special cases.

Findings

Complete classification of Brieskorn polynomials up to arc-analytic equivalence.

Sign of coefficients affects the arc-analytic type.

Extension of previous classifications in two and three variables.

Abstract

It has been recently proved that the arc-analytic type of a singular Brieskorn polynomial determines its exponents. This last result may be seen as a real analogue of a theorem by E. Yoshinaga and M. Suzuki concerning the topological type of complex Brieskorn polynomials. In the real setting it is natural to investigate further by asking how the signs of the coefficients of a Brieskorn polynomial change its arc-analytic type. The aim of the present paper is to answer this question by giving a complete classification of Brieskorn polynomials up to the arc-analytic equivalence. The proof relies on an invariant of this relation whose construction is similar to the one of Denef-Loeser motivic zeta functions. The classification obtained generalizes the one of Koike-Parusi\'nski in the two variable case up to the blow-analytic equivalence and the one of Fichou in the three variable case up to the blow-Nash equivalence.