可分解一元多项式插值
Interpolation by decomposable univariate polynomials
摘要：
通常的一元插值问题是找到一个首一多项式f的次数为n，内插n个给定的值是很好理解的。本文研究了一个变体，其中f是复合的，比如说，一个复合的两个多项式的次数d和e，分别与de=n，因此d+e-1给定值。一些特殊情况很容易解决，对于一般情况，我们构造了它和一个特殊情况之间的同伦。我们计算的几何解决方案的代数曲线提出这个同伦，这也提供了一个答案的插值任务。计算时间是多项式的几何数据，如次数，这条曲线。结果是，对于几乎所有的输入，存在可分解的插值多项式。
英文摘要：
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.
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