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A Homogeneous Nullstellensatz for Joint Invariant Subspaces

Sizhuo Yan, Jianting Yang, Lihong Zhi

Abstract

Jurij Volčič 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.

A Homogeneous Nullstellensatz for Joint Invariant Subspaces

Abstract

Jurij Volčič conjectured that a noncommutative polynomial belongs to the unital -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 . 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.
Paper Structure (6 sections, 10 theorems, 41 equations, 2 algorithms)

This paper contains 6 sections, 10 theorems, 41 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. True zero set
  3. Directional zero set
  4. Homogeneous Nullstellensatz for Joint Invariant Subspaces
  5. Non-homogeneous Case
  6. Further Discussion

Key Result

Theorem 1.1

salomon2018algebras For $f_1,\ldots,f_l,\, g \in \mathbb{K}\langle \underline{x}\rangle$, suppose $f_1,\ldots,f_l$ be homogeneous, the following are equivalent:

Theorems & Definitions (17)

  • Theorem 1.1: Salomon-Shalit-Shamovich
  • Theorem 1.2: Bergman
  • Conjecture 1.3
  • Proposition 1.4
  • Theorem 1.5: Homogeneous Nullstellensatz for Joint Invariant Subspaces
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • proof : proof of Theorem \ref{['thm:homo-case']}
  • Theorem 3.1
  • ...and 7 more