math.RA — Rings and Algebras
The rank-one Dixmier conjecture for elements of mass at most six
Let K be a field of characteristic zero and let P,Q in A_1(K) satisfy [Q,P]=1. We prove that P,Q generate A_1(K) if either has mass at most six, meaning at most six nonzero homogeneous components for the grading deg X=1, deg Y=-1, with no degree bound on either element. This extends the four-component theorem of Guccione, Guccione and Valqui. The proof combines their Newton-polygon restrictions with a classification of polynomial pairs r,f satisfying an associated scalar differential equation under the condition that r^2 has at most six monomial terms. The scalar argument uses a two-dimensional Vandermonde kernel, Rolle's theorem, and a sign and parity obstruction. The scalar classification and the mass bound reduce the crossing cases to a binomial-power face. We exclude that face using at most one Newton cut, without a mass bound on the transformed pair. We also exclude a family of pure-power crossing faces with prime outer exponent, without a mass bound. A seven-term scalar solution shows that the scalar classification is sharp, although its associated Weyl face is excluded by the pure-power theorem. Theorem 1.1 is formalized in Lean 4 and registered as Palomar entry PALOMAR-2026-10-05-000006, version 2.