L'vov-Kaplansky Conjecture

The L'vov-Kaplansky conjecture asserts that the image of every homogeneous multilinear noncommutative polynomial on the algebra of matrices over a field is a -vector space, for all positive integers and . Here the polynomial has polynomial degree and the form

where is the symmetric group, , and each variable occurs exactly once in each product. The factors need not commute. The assertion concerns the set of values with , rather than only its vector space span (Kanel-Belov et al. 2012, 2020).

Equivalently, the conjecture predicts that the image is the zero space, the space of scalar matrices , the trace-zero space , or , where is the identity matrix and trace means matrix trace. The conjecture is established for over an algebraically closed field (Kanel-Belov et al. 2012).

Tang (2026a) claims the conjecture for degrees three and four over every field, including finite fields, and every . Tang (2026b) claims that for each fixed there is an integer bound , independent of the field and the coefficients, such that for every nonzero polynomial of the displayed form has image either or . The first case is asserted to occur exactly when every cyclic coefficient sum vanishes. These sums are obtained by grouping the words into classes under cyclic permutations and adding the coefficients within each class. These claims leave the unrestricted problem in arbitrary polynomial degree and matrix size open.

For algebraically closed fields of characteristic zero, Tang (2026c) claims degrees five and six, and Pan (2026) claims degrees four through six, in every matrix size. Tang (2026abc) credits GPT-6 Pro with developing the results and Codex with reviewing the proofs. Tang's two Hexagon papers report partial human understanding and plan further automated review. Independent specialist verification of these new claims has not been reported.