Evaluation and interpolation over multivariate skew polynomial rings

TL;DR

This paper extends evaluation and interpolation techniques from univariate to multivariate skew polynomial rings over division rings, providing new methods for zero set characterization and Lagrange interpolation.

Contribution

It introduces a construction of multivariate skew polynomial rings as quotients of free rings and extends key concepts like P-closed sets and skew Vandermonde matrices to the multivariate case.

Findings

Descriptions of zero sets of multivariate skew polynomials

Development of Lagrange interpolation methods for these polynomials

Extension of P-independence and P-bases to multivariate skew polynomial rings

Abstract

The concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials, with coefficients over division rings. Iterated skew polynomial rings are in general not suitable for this purpose. Instead, multivariate skew polynomial rings are constructed in this work as follows: First, free multivariate skew polynomial rings are defined, where multiplication is additive on degrees and restricts to concatenation for monomials. This allows to define the evaluation of any skew polynomial at any point by unique remainder division. Multivariate skew polynomial rings are then defined as the quotient of the free ring by (two-sided) ideals that vanish at every point. The main objectives and results of this work are descriptions of the sets of zeros of these multivariate skew polynomials, the families of functions that such skew polynomials define, and how to perform Lagrange interpolation with them. To obtain these descriptions, the existing concepts of P-closed sets, P-independence, P-bases (which are shown to form a matroid) and skew Vandermonde matrices are extended from the univariate case to the multivariate one.