A Computational Approach to Examining Dixmier Conjecture in a Specific Case: Online First: 04/05/2026

Van Hoang Dinh1, Van Minh Man Nguyen2
1Ho Chi Minh City University of Technology and Engineering, Vietnam
2Mahidol University (MUSC), Bangkok 10400, Thailand

Tóm tắt

Dixmier Conjecture on Weyl algebra is one of the central open problems in the field of Lie theory and non-commutative algebra. In this paper, by using the computer algebra system (Maple) we examine a particular instance of this conjecture involving two polynomial generators of relatively low degrees. In parallel, we also study a conjecture introduced in 1997 by Professor Nguyen Huu Anh, which shares deep structural similarities with Dixmier conjecture. Our research reveals a logical relationship between the two conjectures. From a computational perspective, we develop a computer program to systematically construct and analyze all polynomial pairs of degrees (6,9) whose Lie products are constants, thereby confirming the validity of the Dixmier Conjecture in this specific case. Our results contribute to a deeper understanding of the computational and theoretical boundaries of Dixmier conjecture and other related problems in non-commutative algebra.

Từ khóa

#Algebra #Computer Algebra #Lie Algebra #Dixmier Conjecture #Maple software

Tài liệu tham khảo

J. Dixmier, “Sur les algèbres de Weyl,” Bull. Soc. Math. France, vol. 96, pp. 209–242, 1968.

A. Joseph, “The Weyl algebra—semisimple and nilpotent elements,” Amer. J. Math., vol. 97, no. 3, pp. 597–615, 1975.

V. V. Bavula, “Dixmier’s Problem 5 for the Weyl Algebra,” J. Algebra, vol. 283, no. 2, pp. 604–621, 2005.

V. V. Bavula and V. Levandovskyy, “A remark on the Dixmier Conjecture,” Canad. Math. Bull., vol. 63, no. 1, pp. 6–12, 2020.

V. Moskowicz, “About Dixmier’s Conjecture,” J. Algebra Appl., vol. 14, no. 10, 2015.

V. Moskowicz and C. Valqui, “The starred Dixmier Conjecture for A₁,” Comm. Algebra, vol. 43, no. 8, pp. 3073–3082, 2015.

J. A. Guccione, J. J. Guccione, C. Valqui, and M. A. Zubimendi, “The Dixmier Conjecture and the shape of possible counterexamples,” J. Algebra, vol. 399, pp. 581–633, 2014.

G. Han and B. Tan, “Some progress on Dixmier Conjecture for A₁,” Comm. Algebra, vol. 52, no. 5, 2024.

K. Adjamagbo and A. R. P. van den Essen, “A proof of the equivalence of the Dixmier, Jacobian and Poisson Conjectures,” Acta Math. Vietnam., vol. 32, no. 3, pp. 15–23, 2007.

A. Belov-Kanel and M. Kontsevich, “The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture,” Moscow Math. J., vol. 7, no. 2, pp. 209–218, 2007.

Y. Tsuchimoto, “Endomorphisms of Weyl algebra and p-curvatures,” Osaka J. Math., vol. 42, no. 2, pp. 435–452, 2005.

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