Multilinear 3-scalar polynomials on 3×3 matrices: structure, reduction to the reducible locus, and non-existence in degree at most seven
Year: 2026
Type: preprint
Abstract: Kanel-Belov, Malev and Rowen reduced the L’vov–Kaplansky conjecture for 3×3 matrices over an algebraically closed field to six possible images. They asked whether a multilinear polynomial can take only 3-scalar values, that is, matrices with eigenvalues (β, βε, βε²) where ε is a primitive cube root of unity. A positive answer would disprove the conjecture. We work over algebraically closed fields of characteristic zero and prove the following. (i) Such a polynomial is an identity of M₂(K), has coefficient sum zero, and vanishes on a maximal parabolic subalgebra of M₃(K). (ii) If one exists, then one exists that vanishes on every tuple with a common invariant proper subspace. (iii) All of its values on oriented 3-cycles of matrix units lie on a single eigenvalue line. Under the hypothesis of (ii), these values vanish, and so do its values on pairs of 2-cycles sharing a vertex. The proofs use the fact that generic matrices have no zero divisors, together with exact coefficient relations extracted from the identity tr p² = 0. Combining these results with exact modular rank certificates, lifted to characteristic zero by a lattice argument, we show that no multilinear 3-scalar polynomial of degree at most seven exists. The degree-seven certificate is a rank computation of size 14,196. The remaining step for all degrees is isolated as a single statement about matrix-unit configurations: every Eulerian configuration of a 3-scalar polynomial that vanishes on the reducible locus is zero. The LaTeX source and the scripts that produce the certificates are included.
Author: Chuang, William Huanshan
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225467622