Volčič’s Rank Nullstellensatz Conjecture

Let $f_1,\ldots,f_l,g\in\mathbb{F}\langle x_1,\ldots,x_d\rangle$ be noncommutative polynomials. The conjecture asserts that $g$ belongs to the two-sided ideal $(f_1,\ldots,f_l)$ if and only if there exists a constant $C>0$ such that, for every matrix size $n\geq 1$ and every tuple $X=(X_1,\ldots,X_d)\in M_n(\mathbb{F})^d$ , one has $\operatorname{rank} g(X)\leq C\max_{1\leq i\leq l}\operatorname{rank} f_i(X)$ , where $f_i(X)$ and $g(X)$ denote the matrix evaluations.

References

Primary source

arXiv

Additional references

On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials — arXiv — Sizhuo Yan, Jianting Yang

noncommutative-algebra

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed solved

The conjecture is true for homogeneous inputs but false in general, with a reported counterexample showing exactly where it fails.

Volčič’s conjecture proposed an equivalence between an invariant-subspace condition and algebra membership for noncommutative polynomials. The latest work separates the valid homogeneous case from the false unrestricted formulation.

Known results

Homogeneous generators: the equivalence is proved for every matrix size (Yan and Yang, 2026).

October 2026 counterexample

Yan and Yang report that the unrestricted conjecture is false, while the homogeneous version holds; the newer report describes an explicit nonhomogeneous counterexample. This is a specialist preprint result and remains unverified here.

Current status (as of October 2026): The homogeneous case is proved and the unrestricted conjecture is reported false, but the new counterexample and its claimed resolution remain unverified.

Sources

No solutions have been posted yet.