Czechoslovak Mathematical Journal, first online, pp. 1-8


On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun

Received January 26, 2018.   Published online May 24, 2019.

Abstract:  We show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda=(\partial_x-\Phi(\partial_y))\partial_y$ and all polynomials $P(x,y)$, where $\Phi(t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.
Keywords:  Jacobian conjecture; generalized vanishing conjecture; differential operator
Classification MSC:  14R15, 13N15
DOI:  10.21136/CMJ.2019.0049-18

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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] H. Bass, E. H. Connell, 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
[3] A. Belov-Kanel, M. Kontsevich: The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Mosc. Math. J. 7 (2007), 209-218. DOI 10.17323/1609-4514-2007-7-2-209-218 | MR 2337879 | Zbl 1128.16014
[4] M. de Bondt: A few remarks on the generalized vanishing conjecture. Arch. Math. 100 (2013), 533-538. DOI 10.1007/s00013-013-0523-2 | MR 3069106 | Zbl 1273.13046
[5] M. de Bondt, A. van den Essen: Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. J. Pure Appl. Algebra 193 (2004), 61-70. DOI 10.1016/j.jpaa.2004.03.003 | MR 2076378 | Zbl 1054.14083
[6] M. de Bondt, A. van den Essen: A reduction of the Jacobian conjecture to the symmetric case. Proc. Am. Math. Soc. 133 (2005), 2201-2205. DOI 10.1090/S0002-9939-05-07570-2 | MR 2138860 | Zbl 1073.14077
[7] M. de Bondt, A. van den Essen: Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. II. J. Pure Appl. Algebra 196 (2005), 135-148. DOI 10.1016/j.jpaa.2004.08.030 | MR 2110519 | Zbl 1077.14092
[8] D. Liu, X. Sun: Images of higher-order differential operators of polynomial algebras. Bull. Aust. Math. Soc. 96 (2017), 205-211. DOI 10.1017/S0004972717000454 | MR 3703902 | Zbl 1390.13078
[9] O. Mathieu: Some conjectures about invariant theory and their applications. Algèbre non commutative, groupes quantiques et invariants (Alev, J. et al., eds.). Sémin. Congr. 2, Société Mathématique de France, Paris (1995), 263-279. MR 1601155 | Zbl 0889.22008
[10] G. Meng: Legendre transform, Hessian conjecture and tree formula. Appl. Math. Lett. 19 (2006), 503-510. DOI 10.1016/j.aml.2005.07.006 | MR 2221506 | Zbl 1132.14340
[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] Y. Tsuchimoto: Endomorphisms of Weyl algebra and $p$-curvatures. Osaka J. Math. 42 (2005), 435-452. MR 2147727 | Zbl 1105.16024
[13] 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
[14] 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 06867612
[15] 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
[16] 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
[17] A. van den Essen, W. Zhao: Two results on homogeneous Hessian nilpotent polynomials. J. Pure Appl. Algebra 212 (2008), 2190-2193. DOI 10.1016/j.jpaa.2008.01.005 | MR 2418165 | Zbl 1147.14033
[18] 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
[19] W. Zhao: A vanishing conjecture on differential operators with constant coefficients. Acta Math. Vietnam. 32 (2007), 259-286. MR 2368014 | Zbl 1139.14303
[20] W. Zhao: Hessian nilpotent polynomials and the Jacobian conjecture. Trans. Am. Math. Soc. 359 (2007), 249-274. DOI 10.1090/S0002-9947-06-03898-0 | MR 2247890 | Zbl 1109.14041
[21] 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

Affiliations:   Zhenzhen Feng, Xiaosong Sun (corresponding author), School of Mathematics, Jilin University, Qianjin Street, Changchun 130012, China, e-mail: fengzz13@mails.jlu.edu.cn, sunxs@jlu.edu.cn


 
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