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

Xiaosong Sun; Beini Wang

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 599-610
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Sun, Xiaosong, and Wang, Beini. "Images of locally nilpotent derivations of bivariate polynomial algebras over a domain." Czechoslovak Mathematical Journal 74.2 (2024): 599-610. <http://eudml.org/doc/299587>.

@article{Sun2024,
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.},
author = {Sun, Xiaosong, Wang, Beini},
journal = {Czechoslovak Mathematical Journal},
keywords = {locally nilpotent derivation; Jacobian conjecture; LND conjecture; Mathieu-Zhao subspace},
language = {eng},
number = {2},
pages = {599-610},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Images of locally nilpotent derivations of bivariate polynomial algebras over a domain},
url = {http://eudml.org/doc/299587},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Sun, Xiaosong
AU - Wang, Beini
TI - Images of locally nilpotent derivations of bivariate polynomial algebras over a domain
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 599
EP - 610
AB - 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.
LA - eng
KW - locally nilpotent derivation; Jacobian conjecture; LND conjecture; Mathieu-Zhao subspace
UR - http://eudml.org/doc/299587
ER -

References

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  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. (2007) Zbl1137.14046MR2368008
  2. Atiyah, M. F., Macdonald, I. G., 10.1201/9780429493621, Addison-Wesley, Reading (1969). (1969) Zbl0175.03601MR0242802DOI10.1201/9780429493621
  3. Bass, H., Connel, E. H., Wright, D., 10.1090/S0273-0979-1982-15032-7, Bull. Am. Math. Soc., New Ser. 7 (1982), 287-330. (1982) Zbl0539.13012MR0663785DOI10.1090/S0273-0979-1982-15032-7
  4. Belov-Kanel, A., Kontsevich, M., 10.17323/1609-4514-2007-7-2-209-218, Mosc. Math. J. 7 (2007), 209-218. (2007) Zbl1128.16014MR2337879DOI10.17323/1609-4514-2007-7-2-209-218
  5. Bhatwadekar, S., Dutta, A. K., 10.1090/S0002-9947-97-01946-6, Trans. Am. Math. Soc. 349 (1997), 3303-3319. (1997) Zbl0883.13006MR1422595DOI10.1090/S0002-9947-97-01946-6
  6. Francoise, J. P., Pakovich, F., Yomdin, Y., Zhao, W., 10.1016/j.bulsci.2010.06.002, Bull. Sci. Math. 135 (2011), 10-32. (2011) Zbl1217.44008MR2764951DOI10.1016/j.bulsci.2010.06.002
  7. Freudenburg, G., 10.1007/978-3-662-55350-3, Encyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic Transformation Groups 7. Springer, Berlin (2017). (2017) Zbl1391.13001MR3700208DOI10.1007/978-3-662-55350-3
  8. Liu, D., Sun, X., 10.1017/S0004972719000546, Bull. Aust. Math. Soc. 101 (2020), 71-79. (2020) Zbl1430.14113MR4052910DOI10.1017/S0004972719000546
  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. (1997) Zbl0889.22008MR1601155
  10. Shestakov, I. P., Umirbaev, U. U., 10.1090/S0894-0347-03-00438-7, J. Am. Math. Soc. 17 (2004), 181-196. (2004) Zbl1044.17014MR2015333DOI10.1090/S0894-0347-03-00438-7
  11. Sun, X., 10.1016/j.jalgebra.2017.09.020, J. Algebra 492 (2017), 414-418. (2017) Zbl1386.14208MR3709158DOI10.1016/j.jalgebra.2017.09.020
  12. Sun, X., Liu, D., 10.1016/j.jalgebra.2020.10.025, J. Algebra 569 (2021), 401-415. (2021) Zbl1451.14172MR4187241DOI10.1016/j.jalgebra.2020.10.025
  13. Sun, X., Wang, B., 10.1017/S000497272300059X, Bull. Aust. Math. Soc. 108 (2023), 412-421. (2023) Zbl07764668MR4665220DOI10.1017/S000497272300059X
  14. Tsuchimoto, Y., Endomorphisms of Weyl algebra and p -curvatures, Osaka J. Math. 42 (2005), 435-452. (2005) Zbl1105.16024MR2147727
  15. Essen, A. van den, 10.1007/978-3-0348-8440-2, Progress in Mathematics 190. Birkhäuser, Basel (2000). (2000) Zbl0962.14037MR1790619DOI10.1007/978-3-0348-8440-2
  16. Essen, A. van den, Sun, X., 10.1016/j.jpaa.2017.12.003, J. Pure Appl. Algebra 222 (2018), 3219-3223. (2018) Zbl1454.13046MR3795641DOI10.1016/j.jpaa.2017.12.003
  17. Essen, A. van den, Willems, R., Zhao, W., 10.1016/j.jpaa.2014.12.024, J. Pure Appl. Algebra 219 (2015), 3847-3861. (2015) Zbl1317.33007MR3335985DOI10.1016/j.jpaa.2014.12.024
  18. Essen, A. van den, Wright, D., Zhao, W., 10.1016/j.jpaa.2010.12.002, J. Pure Appl. Algebra 215 (2011), 2130-2134. (2011) Zbl1229.13022MR2786603DOI10.1016/j.jpaa.2010.12.002
  19. Essen, A. van den, Wright, D., Zhao, W., 10.1016/j.jalgebra.2011.04.036, J. Algebra 340 (2011), 211-224. (2011) Zbl1235.14057MR2813570DOI10.1016/j.jalgebra.2011.04.036
  20. Wright, D., The Jacobian conjecture as a problem in combinatorics, Affine Algebraic Geometry Osaka University Press, Osaka (2007), 483-503. (2007) Zbl1129.14087MR2330486
  21. Zhao, W., 10.1016/j.jpaa.2009.10.007, J. Pure Appl. Algebra 214 (2010), 1200-1216. (2010) Zbl1205.33017MR2586998DOI10.1016/j.jpaa.2009.10.007
  22. Zhao, W., 10.1016/j.jalgebra.2010.04.022, J. Algebra 324 (2010), 231-247. (2010) Zbl1197.14064MR2651354DOI10.1016/j.jalgebra.2010.04.022
  23. Zhao, W., 10.1016/j.jalgebra.2011.09.036, J. Algebra 350 (2012), 245-272. (2012) Zbl1255.16018MR2859886DOI10.1016/j.jalgebra.2011.09.036
  24. Zhao, W., 10.1142/S0219199717500560, Commun. Contemp. Math. 20 (2018), Article ID 1750056, 25 pages. (2018) Zbl1476.16004MR3810636DOI10.1142/S0219199717500560

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