Images of locally nilpotent derivations of bivariate polynomial algebras over a domain

Sun, Xiaosong ; Wang, Beini

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 599-610

Abstract (VO)

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.

Détail

DOI : 10.21136/CMJ.2024.0008-24

Classification : 13N15, 14R10
Keywords: locally nilpotent derivation; Jacobian conjecture; LND conjecture; Mathieu-Zhao subspace

Bibliographie

[1] Adjamagbo, P. K., Essen, A. van den: A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures. Acta Math. Vietnam. 32 (2007), 205-214. | MR | JFM

[2] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969). | DOI | MR | JFM

[3] Bass, H., Connel, E. H., Wright, D.: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc., New Ser. 7 (1982), 287-330. | DOI | MR | JFM

[4] Belov-Kanel, A., Kontsevich, M.: The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Mosc. Math. J. 7 (2007), 209-218. | DOI | MR | JFM

[5] Bhatwadekar, S., Dutta, A. K.: Kernel of locally nilpotent $R$-derivations of $R[X,Y]$. Trans. Am. Math. Soc. 349 (1997), 3303-3319. | DOI | MR | JFM

[6] Francoise, J. P., Pakovich, F., Yomdin, Y., Zhao, W.: Moment vanishing problem and positivity: Some examples. Bull. Sci. Math. 135 (2011), 10-32. | DOI | MR | JFM

[7] Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic Transformation Groups 7. Springer, Berlin (2017). | DOI | MR | JFM

[8] Liu, D., Sun, X.: The factorial conjecture and images of locally nilpotent derivations. Bull. Aust. Math. Soc. 101 (2020), 71-79. | DOI | MR | JFM

[9] Mathieu, O.: 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 | JFM

[10] Shestakov, I. P., Umirbaev, U. U.: Poisson brackets and two-generated subalgebras of rings of polynomials. J. Am. Math. Soc. 17 (2004), 181-196. | DOI | MR | JFM

[11] Sun, X.: Images of derivations of polynomial algebras with divergence zero. J. Algebra 492 (2017), 414-418. | DOI | MR | JFM

[12] Sun, X., Liu, D.: Images of locally nilpotent derivations of polynomial algebras in three variables. J. Algebra 569 (2021), 401-415. | DOI | MR | JFM

[13] Sun, X., Wang, B.: On the LND conjecture. Bull. Aust. Math. Soc. 108 (2023), 412-421. | DOI | MR | JFM

[14] Tsuchimoto, Y.: Endomorphisms of Weyl algebra and $p$-curvatures. Osaka J. Math. 42 (2005), 435-452. | MR | JFM

[15] Essen, A. van den: Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics 190. Birkhäuser, Basel (2000). | DOI | MR | JFM

[16] Essen, A. van den, Sun, X.: Monomial preserving derivations and Mathieu-Zhao subspaces. J. Pure Appl. Algebra 222 (2018), 3219-3223. | DOI | MR | JFM

[17] Essen, A. van den, Willems, R., Zhao, W.: Some results on the vanishing conjecture of differential operators with constant coefficients. J. Pure Appl. Algebra 219 (2015), 3847-3861. | DOI | MR | JFM

[18] Essen, A. van den, Wright, D., Zhao, W.: Images of locally finite derivations of polynomial algebras in two variables. J. Pure Appl. Algebra 215 (2011), 2130-2134. | DOI | MR | JFM

[19] Essen, A. van den, Wright, D., Zhao, W.: On the image conjecture. J. Algebra 340 (2011), 211-224. | DOI | MR | JFM

[20] Wright, D.: The Jacobian conjecture as a problem in combinatorics. Affine Algebraic Geometry Osaka University Press, Osaka (2007), 483-503. | MR | JFM

[21] Zhao, W.: Generalization of the image conjecture and the Mathieu conjecture. J. Pure Appl. Algebra 214 (2010), 1200-1216. | DOI | MR | JFM

[22] Zhao, W.: Images of commuting differential operators of order one with constant leading coefficients. J. Algebra 324 (2010), 231-247. | DOI | MR | JFM

[23] Zhao, W.: Mathieu subspaces of associative algebras. J. Algebra 350 (2012), 245-272. | DOI | MR | JFM

[24] Zhao, W.: 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 | MR | JFM