Kippenhahn’s conjecture

For Hermitian matrices $A_{1},A_{2}\in M_{n}(\mathbb{C})$ , define $p(z_{0},z_{1},z_{2})=\det\!\left(z_{0}I+z_{1}A_{1}+z_{2}A_{2}\right)$ . Kippenhahn's conjecture asserts that if there is a nonconstant polynomial $q\in\mathbb{C}[z_{0},z_{1},z_{2}]$ such that $q^{2}\mid p$ , then $A_{1}$ and $A_{2}$ are simultaneously reducible; that is, there exists a subspace $V$ with $\{0\}\subsetneq V\subsetneq\mathbb{C}^{n}$ such that $A_{1}V\subseteq V$ and $A_{2}V\subseteq V$ . The conjecture is false in this generality; the cited 2026 preprint claims a complete characterization of the remaining reducibility problem.

Progress summary

Solved

The original conjecture is known to fail in general, but an August 2026 preprint claims to give a complete characterization that resolves the remaining question.

Kippenhahn proposed the conjecture in 1951, linking repeated factors in characteristic polynomials of Hermitian matrix tuples to simultaneous reducibility. Its broad original form is false, but the precise reducibility problem remains the subject of current work.

Known results

Kippenhahn verified special cases where the relevant minimal polynomial has degree $1$ or $2$ .
Laffey gave a counterexample in the 1980s; later work reports versions in orders $8$ and higher.
Li, Spitkovsky, and Shukla (1998) disproved a more general form for $n=6$ .
A 2017 paper constructed a one-parameter family of counterexamples for Hermitian matrices of order $n \ge 8$ .

August 2026 claimed resolution

Stessin and Yang claim that spectral index, spectral stability, and characteristic graphs yield necessary and sufficient conditions and “completely settle” Kippenhahn’s conjecture, including the Hermitian-pair criterion involving $\det(z_{0}I+z_{1}A_{1}+z_{2}A_{2})$ . This is an unrefereed preprint claim with no supplied independent verification.

Current status (as of August 2026): The broad conjecture is disproved by counterexamples, while a new preprint claims to settle the refined general problem; that claimed resolution remains unverified.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Reducibility of linear representations, free ideals, and Kippenhahn's conjecture — arXiv — Michael Stessin, Rongwei Yang

4 days ago