Zeros with multiplicity, Hasse derivatives and linear factors of general skew polynomials

TL;DR

This paper investigates the multiplicities of zeros in skew polynomials, compares two definitions, and extends classical results like bounds on zeros and Hermite interpolation to the non-commutative setting using Hasse derivatives.

Contribution

It introduces a new perspective on zero multiplicities in skew polynomials, analyzes P-independence behavior, and extends classical commutative polynomial results to skew polynomials.

Findings

P-independence behaves naturally under the second zero multiplicity definition.

An upper bound on the number of zeros (with multiplicities) by the polynomial degree is established.

Equivalence of P-independence, Hermite interpolation, and invertibility of skew Vandermonde matrices is shown.

Abstract

In this work, multiplicities of zeros of skew polynomials are studied. Two distinct definitions are considered: First, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if $ (x-a)^r $ divides $ F $ on the right; second, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if some skew polynomial $ P = (x-a_r) \cdots (x-a_2) (x-a_1) $, having $ a_1 = a $ as its only right zero, divides $ F $ on the right. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy's concept of P-independence does not behave naturally, whereas a union theorem still holds. In contrast, we show that P-independence behaves naturally for the second notion of multiplicities. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include: (1) The upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and (2) The equivalence of P-independence, Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives).