Matrix analogue of the “all points” radical characterization

Determine whether an analogue of the quaternionic “all points” characterization of radicals holds for matrix polynomials: specifically, whether for the ring of matrix polynomials M_n(k[x1,...,xd]) over an algebraically closed field k and any left ideal I, there exists a characterization of the radical √I paralleling the quaternionic result that, for H[x1,...,xd], √I equals the set of all quaternionic polynomials that vanish at every point a ∈ H^d (not only central points) at which all elements of I vanish.

Background

In the quaternionic setting H[x1,...,xd], central points are used to define evaluation ideals that are left ideals, and the radical of a left ideal I is classically characterized via vanishing on central points. Recent work (e.g., [7], [2], [3]) shows a stronger description: √I consists of all quaternionic polynomials that vanish at all points in H^d—without restricting to central points—whenever those points are common zeros of I.

For matrix polynomials M_n(k[x1,...,xd]) (with k algebraically closed), the paper develops a comprehensive Nullstellensatz framework using directional points (pairs (a,v) with a ∈ k^d and nonzero v ∈ k^n), and defines the radical √I via annihilation of all directional points annihilated by I. The authors explicitly raise the question of whether there is a parallel to the quaternionic “all points” description in the matrix setting.