Equivalência semialgébrica de Lipschitz de funções polinomiais

Abstract

We show how to determine, under fairly general conditions, whether two given β-quasi-homogeneous polynomials in two variables, with real coefficients, are R-semialgebraically Lipschitz equivalent. Following the strategy used in BIRBRAIR, FERNANDES, and PANAZZOLO (2009), we first show how to determine whether two given real polynomial functions of a single variable are Lipschitz equivalent by comparing the values and also the multiplicities of the given polynomial functions at their critical points, and then we show how to reduce, under fairly general conditions, the problem of R-semialgebraic Lipschitz equivalence of β-quasihomogeneous polynomials in two variables, with real coefficients, to the problem of Lipschitz equivalence of real polynomial functions of a single variable. As an application of our main results on R-semialgebraic Lipschitz equivalence of β-quasihomogeneous polynomials in two variables, we investigate the properties, in the context of R-semialgebraic Lipschitz equivalence, of a specific family of quasihomogeneous polynomials, which has been used before in HENRY and PARUSINSKI (2004), to show that the bi-Lipschitz equivalence of analytic function germs ( R2, 0) → ( R , 0) admits continuous moduli. As a byproduct, our conclusions show that the R-semialgebraic Lipschitz equivalence of real β-quasihomogeneous polynomials in two variables also admits continuous moduli.