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Symbolic differentiation of factorized polynomials with repeated roots and the identification of their loci

机译:具有重复根的因式多项式的符号微分及其位点的标识

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The mathematical apparatus of algebraic combinatorics is excellently suited to the investigation of symbolic differentiation of analytic functions. In particular, the relationship between the algebraic roots of polynomials and those of their derivatives remains an important topic with wide applications to various branches of pure and applied mathematics. In this paper, a methodology inspired by combinatorial theory is employed to derive analytic expressions for the k-th derivative q (k) of factorized polynomial functions with repeated roots $(x^{xi} - a_{1})^{alpha_{1}} (x^{xi} - a_{2})^{alpha_{2}} cdots (x^{xi} - a_{n})^{alpha_{n}}$ , where ξ∈ℝ∗, a i , α i ∈ℝ and i=1,…,n. It is shown that these derivatives are generating functions for classes of integer sequences whose properties are employed to develop a binary tree algorithm that is suitable for the symbolic evaluation of q (k). Compared to the application of Faá di Bruno’s famous differentiation formula for composite functions and to other existing methods for symbolic differentiation, the algorithm is superior because it does not involve finding integer partitions, which is an NP-complete problem. Mathematical identities that relate this topic to other branches of mathematics (e.g. to statistics via the multinomial distribution and multinomial coefficients) are derived and, in addition, a method for identifying the loci of the roots of polynomial derivatives is outlined. The practical significance of these contributions lies in their applicability to various areas of engineering and physics.
机译:代数组合数学的数学仪器非常适合研究解析函数的符号微分。尤其是,多项式的代数根与它们的导数的代数根之间的关系仍然是一个重要主题,在纯数学和应用数学的各个分支中都有广泛的应用。本文采用一种受组合理论启发的方法来导出具有重复根$(x ^ {xi}-a_ {1})的因式多项式函数的第k个导数q(k)的解析表达式^ {alpha_ {1}}(x ^ {xi}-a_ {2})^ {alpha_ {2}} cdots(x ^ {xi}-a_ {n})^ {alpha_ {n}} $ $,其中ξ ∈ℝ∗ ,ai ,αi ∈ℝ和i = 1,…,n。结果表明,这些导数正在生成整数序列类的函数,这些整数序列的属性被用于开发适合q(k)的符号评估的二叉树算法。与Faádi Bruno著名的复合函数微分公式的应用以及其他现有的符号微分方法相比,该算法具有优越性,因为它不涉及查找整数分区,这是一个NP完全问题。推导了将该主题与其他数学分支相关的数学身份(例如,通过多项式分布和多项式系数的统计数据),此外还概述了一种用于识别多项式导数根的基因座的方法。这些贡献的实际意义在于它们对工程和物理各个领域的适用性。

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