# P2746: Least degree of a two-variable identity for M3(F2) - ID: `P2746` - Reference: `m3-f2-two-variable-pi-degree` - Page: https://theoremdb.org/statements/P2746 - Export scope: built Markdown snapshot. The current public packet may have changed since this build. - Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e - Current Markdown: https://api.theoremdb.org/v1/statements/m3-f2-two-variable-pi-degree?representation=markdown - Record maturity: Prospecting problem with recorded work ## The problem Determine the least integer \(d\) for which there is a nonzero polynomial \(f\in\mathbb F_2\langle X,Y\rangle\) of total degree \(d\) satisfying \(f(A,B)=0\) for every ordered pair \((A,B)\in M_3(\mathbb F_2)^2\). ### Definitions - **Definition.** The free associative algebra F_2 consists of finite F_2-linear combinations of words in the noncommuting letters X and Y. - **Definition.** The total degree of a nonzero polynomial is the maximum word length with nonzero coefficient. Polynomial evaluation substitutes matrices for X and Y and uses matrix multiplication in word order. ### What counts as a solution - Give an explicit nonzero identity f of degree d and verify it on every pair of 3 by 3 matrices over F2. - Certify that the evaluation map is injective in every smaller degree, or provide an equivalent exact lower-bound proof. ## Status **Resolution status:** Open. **Packet-reported status.** As of 2026-08-01, the checked matrix-invariant and PI sources do not determine the least degree of a nonzero two-variable identity for M_3(F_2). The full exact parameter remains unresolved in this packet. [1](#reference-1) [2](#reference-2) **Recorded scope.** Scope statement unavailable. Read the exact record before reusing this result. [Read this exact record and its supporting evidence](/records/m3-f2-two-variable-pi-degree-status-20260801) *This research-packet status has not passed editorial review.* ## Research packet ### Evidence for the packet-reported status **Claim 1 (Current checked status and unresolved remainder).** As of 2026-08-01, the checked matrix-invariant and PI sources do not determine the least degree of a nonzero two-variable identity for M_3(F_2). The full exact parameter remains unresolved in this packet. A dated review on 2026-08-01 checked the structured sources below, exact-parameter searches, nearby formulations, and the current TheoremDB corpus. Neither checked source determines the requested least degree. A complete resolution must exhibit an identity at the claimed degree and certify injectivity of the evaluation map in every smaller degree. ### Background and intake notes Evaluation matrices, conjugacy-orbit reductions, and nullspace certificates can be reused for nearby finite PI questions. The answer separates identities forced by matrix size from identities caused by the small base field. - Original intake status: UNKNOWN as of 2026-07-27. Classical PI theory gives identities for matrix algebras, while the exact least degree under the two-variable and F2 restrictions was not located. - A 2026-07-27 search checked finite-matrix invariant literature and searches combining M_3(F_2), two generators, and minimal polynomial-identity degree. No exact value was located. - The choice M_3(F_2) is the first 3 by 3 matrix algebra over the smallest field. Restricting to two variables turns the question into an exact finite evaluation-kernel computation without importing the six independent variables of the standard identity. - At each degree, evaluate the word basis on all matrix pairs or use orbit representatives under simultaneous conjugation. A nullspace vector is an explicit identity, while full column rank is a lower-degree certificate. - Trap: identities obtained by multilinearization can collapse in characteristic two. The submitted polynomial must remain nonzero as an element of the free algebra. - Recorded example: The degree-two polynomial XY+YX is not an identity: for A=E_12 and B=E_21 it evaluates to E_11+E_22. ### Open directions - **Route 1** (reported): Compute the graded evaluation maps on matrix-pair orbit representatives until a kernel appears, then retain a kernel vector and full-rank certificates below it. [1](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `m3-f2-two-variable-pi-degree`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Harm Derksen and Visu Makam, Polynomial degree bounds for matrix semi-invariants, Advances in Mathematics 310 (2017), 44-63; arXiv:1512.03393v4. Matrix semi-invariant degree bounds throughout. https://arxiv.org/abs/1512.03393 - Also cited at Dataset references and 2026-08-01 exact-parameter status search. - Also cited at Editorial research route recorded 2026-08-01. - preprint; reference source; arXiv:1512.03393v4; checked 2026-08-01 - Source use: citation_only - Source used to assess the problem's recorded status. - Supplies nearby degree-bound theory, while it does not determine the least degree of a two-variable identity for M_3(F_2). - Nearby degree-bound theory does not settle this two-variable finite-field identity degree. - Source named by the research packet. 2. Vesselin Drensky, Computing with matrix invariants, Mathematica Balkanica 21 (2007), 141-172; arXiv:math/0506614v2. Survey of computational matrix-invariant and polynomial-identity methods. https://arxiv.org/abs/math/0506614 - Also cited at Computational matrix-invariant and PI methods. - preprint; reference source; arXiv:math/0506614v2; checked 2026-08-01 - Source use: citation_only - Source used to assess the problem's recorded status. - Gives computational context for matrix identities, without tabulating this constrained finite-field parameter. - Provides method context without an exact value for this parameter.