可分解单变量多项式插值法

IF 1.8 2区 数学 Q1 MATHEMATICS
Joachim von zur Gathen , Guillermo Matera
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引用次数: 0

摘要

通常的单变量插值问题是寻找一个能插值给定值的度数单项式。本文研究的是一个变式,它要求是复合的,比如说,是两个度分别为 和 的多项式的组成,其中 , 和 为给定值。一些特殊情况很容易求解,对于一般情况,我们在它和特殊情况之间构建了一个同调。我们计算出呈现该同调的代数曲线的 a,这也为插值任务提供了答案。计算时间与该曲线的几何数据(如阶数)成多项式关系。因此,对于几乎所有输入,都存在可分解的插值多项式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Interpolation by decomposable univariate polynomials

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 with d+e1 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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来源期刊
Journal of Complexity
Journal of Complexity 工程技术-计算机:理论方法
CiteScore
3.10
自引率
17.60%
发文量
57
审稿时长
>12 weeks
期刊介绍: The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.
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