Czechoslovak Mathematical Journal, first online, pp. 1-12
Images of locally nilpotent derivations of bivariate polynomial algebras over a domain
Xiaosong Sun, Beini Wang
Received January 4, 2024. Published online May 6, 2024.
Abstract: We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let $R$ be a domain containing a field of characteristic zero. We prove that, when $R$ is a one-dimensional unique factorization domain, the image of any locally nilpotent $R$-derivation of the bivariate polynomial algebra $R[x,y]$ is a Mathieu-Zhao subspace. Moreover, we prove that, when $R$ is a Dedekind domain, the image of a locally nilpotent $R$-derivation of $R[x,y]$ with some additional conditions is a Mathieu-Zhao subspace.
References: [1] P. K. Adjamagbo, A. van den Essen: A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures. Acta Math. Vietnam. 32 (2007), 205-214. MR 2368008 | Zbl 1137.14046
[2] M. F. Atiyah, I. G. Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969). DOI 10.1201/9780429493621 | MR 0242802 | Zbl 0175.03601
[3] H. Bass, E. H. Connel, D. Wright: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc., New Ser. 7 (1982), 287-330. DOI 10.1090/S0273-0979-1982-15032-7 | MR 0663785 | Zbl 0539.13012
[4] A. Belov-Kanel, M. Kontsevich: The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Mosc. Math. J. 7 (2007), 209-218. MR 2337879 | Zbl 1128.16014
[5] S. Bhatwadekar, A. K. Dutta: Kernel of locally nilpotent $R$-derivations of $R[X,Y]$. Trans. Am. Math. Soc. 349 (1997), 3303-3319. DOI 10.1090/S0002-9947-97-01946-6 | MR 1422595 | Zbl 0883.13006
[6] J. P. Francoise, F. Pakovich, Y. Yomdin, W. Zhao: Moment vanishing problem and positivity: Some examples. Bull. Sci. Math. 135 (2011), 10-32. DOI 10.1016/j.bulsci.2010.06.002 | MR 2764951 | Zbl 1217.44008
[7] G. Freudenburg: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic Transformation Groups 7. Springer, Berlin (2017). DOI 10.1007/978-3-662-55350-3 | MR 3700208 | Zbl 1391.13001
[8] D. Liu, X. Sun: The factorial conjecture and images of locally nilpotent derivations. Bull. Aust. Math. Soc. 101 (2020), 71-79. DOI 10.1017/S0004972719000546 | MR 4052910 | Zbl 1430.14113
[9] O. Mathieu: Some conjectures about invariant theory and their applications. Algèbre noncommutative, groupes quantiques et invariants. Séminaires et Congrès 2. Société Mathématique de France, Paris (1997), 263-279. MR 1601155 | Zbl 0889.22008
[10] I. P. Shestakov, U. U. Umirbaev: Poisson brackets and two-generated subalgebras of rings of polynomials. J. Am. Math. Soc. 17 (2004), 181-196. DOI 10.1090/S0894-0347-03-00438-7 | MR 2015333 | Zbl 1044.17014
[11] X. Sun: Images of derivations of polynomial algebras with divergence zero. J. Algebra 492 (2017), 414-418. DOI 10.1016/j.jalgebra.2017.09.020 | MR 3709158 | Zbl 1386.14208
[12] X. Sun, D. Liu: Images of locally nilpotent derivations of polynomial algebras in three variables. J. Algebra 569 (2021), 401-415. DOI 10.1016/j.jalgebra.2020.10.025 | MR 4187241 | Zbl 1451.14172
[13] X. Sun, B. Wang: On the LND conjecture. Bull. Aust. Math. Soc. 108 (2023), 412-421. DOI 10.1017/S000497272300059X | MR 4665220 | Zbl 07764668
[14] Y. Tsuchimoto: Endomorphisms of Weyl algebra and $p$-curvatures. Osaka J. Math. 42 (2005), 435-452. MR 2147727 | Zbl 1105.16024
[15] A. van den Essen: Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics 190. Birkhäuser, Basel (2000). DOI 10.1007/978-3-0348-8440-2 | MR 1790619 | Zbl 0962.14037
[16] A. van den Essen, X. Sun: Monomial preserving derivations and Mathieu-Zhao subspaces. J. Pure Appl. Algebra 222 (2018), 3219-3223. DOI 10.1016/j.jpaa.2017.12.003 | MR 3795641 | Zbl 1454.13046
[17] A. van den Essen, R. Willems, W. Zhao: Some results on the vanishing conjecture of differential operators with constant coefficients. J. Pure Appl. Algebra 219 (2015), 3847-3861. DOI 10.1016/j.jpaa.2014.12.024 | MR 3335985 | Zbl 1317.33007
[18] A. van den Essen, D. Wright, W. Zhao: Images of locally finite derivations of polynomial algebras in two variables. J. Pure Appl. Algebra 215 (2011), 2130-2134. DOI 10.1016/j.jpaa.2010.12.002 | MR 2786603 | Zbl 1229.13022
[19] A. van den Essen, D. Wright, W. Zhao: On the image conjecture. J. Algebra 340 (2011), 211-224. DOI 10.1016/j.jalgebra.2011.04.036 | MR 2813570 | Zbl 1235.14057
[20] D. Wright: The Jacobian conjecture as a problem in combinatorics. Affine Algebraic Geometry Osaka University Press, Osaka (2007), 483-503. MR 2330486 | Zbl 1129.14087
[21] W. Zhao: Generalization of the image conjecture and the Mathieu conjecture. J. Pure Appl. Algebra 214 (2010), 1200-1216. DOI 10.1016/j.jpaa.2009.10.007 | MR 2586998 | Zbl 1205.33017
[22] W. Zhao: Images of commuting differential operators of order one with constant leading coefficients. J. Algebra 324 (2010), 231-247. DOI 10.1016/j.jalgebra.2010.04.022 | MR 2651354 | Zbl 1197.14064
[23] W. Zhao: Mathieu subspaces of associative algebras. J. Algebra 350 (2012), 245-272. DOI 10.1016/j.jalgebra.2011.09.036 | MR 2859886 | Zbl 1255.16018
[24] W. Zhao: Some open problems on locally finite or locally nilpotent derivations and $\Cal{E}$-derivations. Commun. Contemp. Math. 20 (2018), Article ID 1750056, 25 pages. DOI 10.1142/S0219199717500560 | MR 3810636 | Zbl 1476.16004
Affiliations: Xiaosong Sun (corresponding author), Beini Wang, School of Mathematics, Jilin University, No. 2699 Qianjin Street, Changchun 130012, Jilin Province, P. R. China, e-mail: sunxs@jlu.edu.cn, wangbn21@jlu.edu.cn
