HEXAGONbeta

The rank-one Dixmier conjecture for elements of mass at most six

A mix of human-written and AI-generated textHuman understanding: all partsmath.RA — Rings and Algebras

Contributed by Idris Ali Shaik ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

OpenAI Codex assisted with mathematical exploration, Lean formalization, and manuscript preparation. The author reviewed the AI-assisted material and takes full responsibility for the manuscript.

Formalizations

Tools used

OpenAI
Codex
Lean
Lean

References

  1. J. Dixmier, Sur les algèbres de Weyl , Bulletin de la Société Mathématique de France 96 (1968), 209--242. DOI: 10.24033/bsmf.1667 (https://www.numdam.org/articles/10.24033/bsmf.1667/).DOI
  2. S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two , arXiv:2608.00222v1 (2026), preprint. Versioned record (https://arxiv.org/abs/2608.00222v1).arXiv
  3. T. Tao, A digestion of the Jacobian conjecture counterexample , What's New, 21 July 2026, online exposition. Article (https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/).link
  4. H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse , Bulletin of the American Mathematical Society (N.S.) 7 (1982), no. 2, 287--330. DOI: 10.1090/S0273-0979-1982-15032-7 (https://doi.org/10.1090/S0273-0979-1982-15032-7).DOI
  5. A. Belov-Kanel and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture , Moscow Mathematical Journal 7 (2007), no. 2, 209--218. arXiv:math/0512171v2 (https://arxiv.org/html/math/0512171v2).arXiv
  6. V. V. Bavula and V. Levandovskyy, A remark on the Dixmier Conjecture , Canadian Mathematical Bulletin 63 (2020), 6--12. DOI: 10.4153/S0008439519000122 (https://doi.org/10.4153/S0008439519000122). arXiv:1812.00042v1 (https://arxiv.org/abs/1812.00042v1).arXivDOI
  7. G. Han and B. Tan, Some progress in the Dixmier conjecture for A1A_1 , Communications in Algebra 52 (2024), no. 5, 2033--2051. DOI: 10.1080/00927872.2023.2280189 (https://doi.org/10.1080/00927872.2023.2280189). Result numbering refers to arXiv:2210.00257v1 (https://arxiv.org/html/2210.00257v1).arXivDOI
  8. J. A. Guccione, J. J. Guccione and C. Valqui, Number of homogeneous components of counterexamples to the Dixmier conjecture , Communications in Algebra 53 (2025), no. 3, 1307--1319. DOI: 10.1080/00927872.2024.2407991 (https://doi.org/10.1080/00927872.2024.2407991). Theorem numbering refers to arXiv:2402.11135v1 (https://arxiv.org/html/2402.11135v1).arXivDOI
  9. A. Zheglov, The Conjecture of Dixmier for the first Weyl algebra is true , arXiv:2410.06959v5 (2026), preprint. Versioned record (https://arxiv.org/abs/2410.06959v5).arXiv
  10. I. A. Shaik, The rank-one Dixmier conjecture for elements of mass at most six , Lean 4 formalization (2026), Palomar entry PALOMAR-2026-10-05-000006, version 2, MIT license. Registered formalization (https://palomar-registry.org/entry?id=PALOMAR-2026-10-05-000006&version=2). Versioned source (https://github.com/shaikidris/DixmierMassSix-Palomar/tree/61783d52b6ae44cd2d8d20ad6cb798e7bbbce3ff). Official verification (https://github.com/PalomarRegistry/PalomarSubmission/actions/runs/37353942411).link
Show all 13 referencesShow fewer references
  1. B. Brindza, Remarks on the number of non-zero coefficients of polynomials , Acta Mathematica Academiae Paedagogicae Nyíregyháziensis (N.S.) 17 (2001), no. 2, 77--79. Journal archive (https://www.emis.de/journals/AMAPN/vol17_2/5.html).link
  2. S. Mattarei, Root multiplicities and number of nonzero coefficients of a polynomial , Journal of Algebra and Its Applications 6 (2007), no. 3, 469--475. DOI: 10.1142/S0219498807002338 (https://doi.org/10.1142/S0219498807002338). arXiv:math/0512239v2 (https://arxiv.org/html/math/0512239v2).arXivDOI
  3. J. A. Guccione, J. J. Guccione and C. Valqui, The Dixmier conjecture and the shape of possible counterexamples , Journal of Algebra 399 (2014), 581--633. DOI: 10.1016/j.jalgebra.2013.10.011 (https://doi.org/10.1016/j.jalgebra.2013.10.011). Theorem numbering refers to arXiv:1111.6100v3 (https://arxiv.org/html/1111.6100v3).arXivDOI

Version history

  1. v1Submitted by Idris Ali ShaikInitial depositCurrentOct 15, 2026