Deformations of functions without real critical points

Abstract

Given a real analytic function f(x, y) with one critical point P-0, we Study deformations f(1) of f such that, for any t not equal 0, the analytic function f(t) has no critical points in a neighborhood of P-0. We give explicitly a deformation without real critical points for any function which has only one real branch with characteristic exponents (4, 2q, r).