Constructing separable Arnold snakes of Morse polynomials

Abstract

We give a new and constructive proof of the existence of a large class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and simple and all its critical values are distinct. Thus to any Morse polynomial we can associate an alternating permutation called Arnold snake, given by the relative positions of its critical values. Our main result is an effective construction that can realise any separable alternating permutation as the Arnold snake of a Morse polynomial.