# Constructing separable Arnold snakes of Morse polynomials

### Miruna-Ştefana Sorea

Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

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## 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.

## Cite this article

Miruna-Ştefana Sorea, Constructing separable Arnold snakes of Morse polynomials. Port. Math. 77 (2020), no. 2 pp. 219–260

DOI 10.4171/PM/2050