The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and therefore d+e-1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.

中文翻译：

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通过可分解单变量多项式进行插值

很好地理解了找到内插n个给定值的n次单项多项式f的常见单变量内插问题。本文研究了一个变体，其中f需要合成，也就是说，分别是度为d和e的两个多项式的合成，其中de = n，因此d + e-1为给定值。一些特殊情况很容易解决，对于一般情况，我们在特殊情况下构造同伦。我们计算出呈现该同伦性的代数曲线的几何解，这也为插值任务提供了答案。计算时间是该曲线的几何数据（如度）中的多项式。结果是，对于几乎所有输入，都存在可分解的插值多项​​式。