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

Abstract

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∗, ai, αi∈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 Faa 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.