Arc-analytic roots of analytic functions are Lipschitz

摘要

Let g be an arc-analytic function (i.e., analytic on every analytic arc) and assume that for some integer r the function g(r) is real analytic. We prove that g is locally Lipschitz; even C-1 if r is less than the multiplicity of g(r). We show that the result fails if g(r) is only a C-k, arc-analytic function (even blow-analytic), k is an element of N. We also give an example of a non-Lipschitz arc-analytic solution of a polynomial equation P(x, y) = y(d) + Sigma(i=1)(d) a(i)(x)y(d-i), where a(i) are real analytic functions. [References: 19]