The degree filtration of the Alpöge subalgebra: retargeting cannot lower the Markus–Yamabe dimension below fourteen
Abstract
A recent chain realization theorem of Castañeda, Honorato and Valenzuela-Henríquez turns a polynomial Keller map F of ℝⁿ with a multi-point fiber into a polynomial vector field on ℝ^N, N=Σᵢmax(deg Fᵢ,2)-n, whose Jacobian has spectrum {-1} everywhere and whose singularities are the fiber; fed with Alpöge's three-dimensional counterexample to the Jacobian conjecture it refutes the weak Markus–Yamabe conjecture in dimension 14. The authors close by asking for the minimum of Σᵢmax(deg(Acirc Fcirc B)ᵢ,2) over polynomial automorphisms A,B of ℝ³, every unit removed being one dimension off the counterexample. We settle the target side of that minimisation, which is a question about the degree filtration of the subalgebra generated by the components of F. Among all polynomials of degree at most 8 in the three generators, those of total degree at most 3, 5 and 6 span spaces of dimensions exactly 1, 2 and 3: spanned by 1; by 1 and R; and by 1, R and Q. This does not follow from the numerical semigroup ⟨4,6,7⟩: the three leading forms are algebraically dependent, the generators are not a SAGBI basis, and genuine subduction occurs. The consequence is that Σᵢmax(deg(Acirc F)ᵢ,2) ≥ 4+6+7=17 for every polynomial map A of degree at most 8 whose composite with F is dominant — a class wider than the automorphisms the question asks about — so no change of target coordinates lowers the realized dimension below 14. The certificate behind the filtration is a 162 × 162 integer identity, machine-checked in Lean 4.
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Claim ledger
Stated results
MY1known2026-08-29
The Alpoge map: det J F = -2 as a ring identity in the nine partial derivatives, the same determinant from the formal Jacobian of the sparse representation, and three distinct rational points with common image (-1/4,0,0)
MY2known2026-08-29
The normalized map F = (R/2, Q, P): the expanded components, degree vector (4,6,7), sum max(dᵢ,2) - 3 = 14, and a root of lambda³ - 58 lambda² - (289/2) lambda - 1 in (-1,0) so that -F is not Hurwitz on R³
MY3known2026-08-29
The resolvent cubic 2 P r³ - Q r² + 2 r - R = 0 at r = x/(1+xy), and the source's collision as its totally split fiber r in 0,2,-2
MY4candidate2026-08-29
The degree filtration of the subalgebra Q[R,Q,P] generated by the Alpoge map: among all polynomials of degree at most 8 in the three generators, the elements of total degree at most 3, 5, 6 form spaces of dimension exactly 1, 2, 3, spanned by 1, then 1 and R, then 1, R and Q
MY5candidate2026-08-29
Retargeting cannot beat 17: for every polynomial map A of R³ of degree at most 8 such that 1, A₁(F), A₂(F), A₃(F) are linearly independent, sumᵢ max(deg (A o F)ᵢ, 2) >= 4+6+7 = 17, so the chain realization of arXiv:2608.05392 cannot produce a dimension below 14 from any change of target coordinates
MY6known data2026-08-29
The dimension-fourteen field of arXiv:2608.05392 Theorem 5.1, kernel-checked: J X + I₁4 is nilpotent of index exactly 14 (M¹4 = 0, M¹3 has 33 nonzero entries), the field has degree 7, and the three rational points S₁, S₂, S₃ are distinct singularities
MY7measurement2026-08-29
Measured negative on the source side: of the 966 = 21 x 46 elementary triangular right compositions F o B with B a single-monomial shear of degree 2 or 3 and 46 rational coefficients, none lowers sum max(deg,2) below 17; the best is 18, attained only by B = (x, y - xz/3, z)
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- On 5 August 2026 Castañeda, Honorato and Valenzuela-Henríquez posted arXiv:2608.05392, *The weak Markus–Yamabe conjecture fails in dimension 14*. A C¹ map X: ℝⁿ → ℝⁿ is Hurwitz if every eigenvalue of JX(x) has negative real part; the weak Markus–Yamabe conjecture says a Hurwitz map is injective. The paper's engine is a chain realization theorem: any polynomial Keller map F = I + H of ℝⁿ with component degrees d₁,…,dₙ yields an explicit polynomial vector field on ℝᴺ,
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- 2026-09-07 03:53 UTC
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