Polynomial interpolation over quaternions

TL;DR

This paper explores polynomial interpolation over quaternions, revealing that the nature of solutions varies significantly depending on whether interpolation conditions are on the left, right, or both, highlighting differences from complex polynomial interpolation.

Contribution

It extends interpolation theory to quaternionic polynomials, characterizing solution sets for different types of interpolation conditions and identifying the structure of solutions in non-commutative settings.

Findings

Unique low-degree solutions for same-type conditions

Infinite solutions when both left and right conditions are involved

Solution sets form ideals or quasi-ideals in quaternionic polynomial ring

Abstract

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring ${\mathbb H}[z]$. The problem containing both "left" and "right" interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in ${\mathbb H}[z]$.