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.

中文翻译：

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构建莫尔斯多项式的可分离阿诺德蛇

我们给出了一类特殊的单变量多项式的存在的新的和建设性的证明，其图形具有预先指定的形状。根据定义，莫尔斯多项式函数的所有临界点都是实数且不同的，并且其所有临界值都是不同的。因此，我们可以将其关联到一个交替排列：所谓的阿诺德蛇，由其临界值的相对位置给出。我们将任何可分离的交替排列实现为莫尔斯多项式的阿诺德蛇。