On skew partial derivatives and a Hermite-type interpolation problem

TL;DR

This paper introduces right and left partial derivatives in multivariate skew polynomial rings and applies these concepts to solve a Hermite-type interpolation problem with constructive methods and algorithms.

Contribution

It defines and explores properties of skew partial derivatives and uses them to develop algorithms for solving Hermite-type interpolation problems in skew polynomial rings.

Findings

Defined skew partial derivatives and proved their properties

Developed constructive algorithms for Hermite-type interpolation

Provided solutions for multivariate skew polynomial interpolation

Abstract

Let $\mathcal{R}:=\mathbb{F}[{\bf x};\sigma,\delta]$ be a multivariate skew polynomial ring over a division ring $\mathbb{F}$. In this paper, we introduce the notion of right and left $(\sigma,\delta)$-partial derivatives of polynomials in $\mathcal{R}$ and we prove some of their main properties. As an application of these results, we solve in $\mathcal{R}$ a Hermite-type multivariate skew polynomial interpolation problem. The main technical tools and results used here are of constructive type, showing methods and algorithms to construct a polynomial in $\mathcal{R}$ which satisfies the above Hermite-type interpolation problem and its relative Lagrange-type version.