Constructing Separable Arnold Snakes of Morse Polynomials
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Submission date: 15. Apr. 2019
MSC-Numbers: 14P25, 14P05, 14H20, 14B05, 05C05, 05A05, 14Q05, 26C
Keywords and phrases: Arnold snake, contact tree, separable permutation, Morse polynomial
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We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values are distinct. Thus we can associate to it an alternating permutation: the so-called Arnold snake, given by the relative positions of its critical values. We realise any separable alternating permutation as the Arnold snake of a Morse polynomial.