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Keywords:
Jacobian conjecture; generalized vanishing conjecture; differential operator
Jacobian conjecture; generalized vanishing conjecture; differential operator
Summary:
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.
References:
[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 2368008 | Zbl 1137.14046
[2] Bass, H., Connell, 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 10.1090/S0273-0979-1982-15032-7 | MR 0663785 | Zbl 0539.13012
[3] Belov-Kanel, A., Kontsevich, M.: 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] Bondt, M. de: 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] Bondt, M. de, Essen, A. van den: 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] Bondt, M. de, Essen, A. van den: 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] Bondt, M. de, Essen, A. van den: 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] Liu, D., Sun, X.: 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] Mathieu, O.: Some conjectures about invariant theory and their applications. Algèbre non commutative, groupes quantiques et invariants Alev, J. et al. Sémin. Congr. 2, Société Mathématique de France, Paris (1995), 263-279. MR 1601155 | Zbl 0889.22008
[10] Meng, G.: 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] Sun, X.: 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] Tsuchimoto, Y.: Endomorphisms of Weyl algebra and $p$-curvatures. Osaka J. Math. 42 (2005), 435-452. MR 2147727 | Zbl 1105.16024
[13] Essen, A. van den: 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] Essen, A. van den, Sun, X.: 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] 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 10.1016/j.jpaa.2014.12.024 | MR 3335985 | Zbl 1317.33007
[16] Essen, A. van den, Wright, D., Zhao, W.: On the image conjecture. J. Algebra 340 (2011), 211-224. DOI 10.1016/j.jalgebra.2011.04.036 | MR 2813570 | Zbl 1235.14057
[17] Essen, A. van den, Zhao, W.: 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] Wright, D.: The Jacobian conjecture as a problem in combinatorics. Affine Algebraic Geometry Osaka University Press, Osaka (2007), 483-503. MR 2330486 | Zbl 1129.14087
[19] Zhao, W.: A vanishing conjecture on differential operators with constant coefficients. Acta Math. Vietnam. 32 (2007), 259-286. MR 2368014 | Zbl 1139.14303
[20] Zhao, W.: 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] Zhao, W.: 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
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