Lagrange interpolation over division rings

TL;DR

This paper investigates Lagrange interpolation over division rings, establishing solvability criteria and describing solutions for mixed left and right interpolation conditions, extending previous work on polynomial independence.

Contribution

It provides the first comprehensive solvability criterion for mixed left and right interpolation problems over division rings and characterizes low-degree solutions.

Findings

Established solvability criterion for mixed interpolation problems

Described all low-degree solutions for combined conditions

Extended previous polynomial independence results

Abstract

For a division ring $\mathbb F$, the polynomials $f\in\mathbb F$ can be evaluated "on the left" and "on the right" giving rise to left and right Lagrange interpolation problems. The problems containig interpolation conditions of the same type were considered in \cite{lam1} where the solvability criterion was given in terms of polynomial independence of interpolation nodes. We establish the solvability criterion and describe all solutions of low degree (less than the number of interpolation conditions imposed) for the problem containing both "left" and "right" conditions.