[#P2746] Least degree of a two-variable identity for M3(F2)
Contents
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 \((A,B)\in M_3(\mathbb F_2)^2\).
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Definitions and notation
1Definitions
Definition 1. The free associative algebra F_2<X,Y> consists of finite F_2-linear combinations of words in the noncommuting letters X and Y.
Definition 2. 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.
2What 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.
1Status
Saved packet · August 1, 2026
Packet-reported status (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.[1][2]
This research-packet status has not passed editorial review.
1Packet records
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Notes and companion material
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 1. 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.
How the 2 records connect
ProblemLeast degree of a two-variable identity for M3(F2)
All 1 recorded relations between these records and the problem
2See also
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“Least degree of a two-variable identity for M3(F2).” TheoremDB. P2746. Problem statement; statement text SHA-256 a3c9b7a2359d85ba7f5b2dc7a6b4f1f2ea4f2b7bac7c78c01333809193f2497e. https://theoremdb.org/statement/?ref=P2746
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This problem includes 2 records joined by 1 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. 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. ↗preprint · reference source · arXiv:1512.03393v4 · checked 2026-08-01Source use: citation only.Source used to assess the problem's recorded status.Also cited at Dataset references and 2026-08-01 exact-parameter status search.Also cited at Editorial research route recorded 2026-08-01.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.
- Vesselin Drensky, Computing with matrix invariants, Mathematica Balkanica 21 (2007), 141-172; arXiv:math/0506614v2. Survey of computational matrix-invariant and polynomial-identity methods. ↗preprint · reference source · arXiv:math/0506614v2 · checked 2026-08-01Source use: citation only.Source used to assess the problem's recorded status.Also cited at Computational matrix-invariant and PI methods.Gives computational context for matrix identities, without tabulating this constrained finite-field parameter.Provides method context without an exact value for this parameter.
CC0 exact polynomial-identity problem for the smallest nonscalar matrix algebra over F2 where finite-field effects are substantial.
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