A Homogeneous Nullstellensatz for Joint Invariant Subspaces

TL;DR

This paper proves a homogeneous Nullstellensatz for joint invariant subspaces in noncommutative algebra, confirming a conjecture in the case of homogeneous generators and clarifying its limitations.

Contribution

It establishes the equivalence for homogeneous generators and shows the failure in the general case, settling Jurij Volčič's conjecture.

Findings

Proves the Nullstellensatz for homogeneous generators.

Demonstrates the failure of the statement in the non-homogeneous case.

Settles the conjecture completely.

Abstract

Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the unital $\mathbb{K}$-algebra generated by finitely many noncommutative polynomials if and only if, for matrices of every size, every joint invariant subspace of the evaluations of the generators is also invariant under the evaluation of $g$. In this paper, we establish a homogeneous Nullstellensatz for joint invariant subspaces by proving that this equivalence holds whenever the generators are homogeneous. In contrast, we demonstrate that the statement fails in the general case, thereby settling the conjecture completely.