| Title:
|
On the generalized vanishing conjecture (English) |
| Author:
|
Feng, Zhenzhen |
| Author:
|
Sun, Xiaosong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
1061-1068 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Jacobian conjecture |
| Keyword:
|
generalized vanishing conjecture |
| Keyword:
|
differential operator |
| MSC:
|
13N15 |
| MSC:
|
14R15 |
| idZBL:
|
07144875 |
| idMR:
|
MR4039620 |
| DOI:
|
10.21136/CMJ.2019.0049-18 |
| . |
| Date available:
|
2019-11-28T08:51:01Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147914 |
| . |
| Reference:
|
[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. Zbl 1137.14046, MR 2368008 |
| Reference:
|
[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. Zbl 0539.13012, MR 0663785, 10.1090/S0273-0979-1982-15032-7 |
| Reference:
|
[3] Belov-Kanel, A., Kontsevich, M.: The Jacobian conjecture is stably equivalent to the Dixmier conjecture.Mosc. Math. J. 7 (2007), 209-218. Zbl 1128.16014, MR 2337879, 10.17323/1609-4514-2007-7-2-209-218 |
| Reference:
|
[4] Bondt, M. de: A few remarks on the generalized vanishing conjecture.Arch. Math. 100 (2013), 533-538. Zbl 1273.13046, MR 3069106, 10.1007/s00013-013-0523-2 |
| Reference:
|
[5] Bondt, M. de, Essen, A. van den: Nilpotent symmetric Jacobian matrices and the Jacobian conjecture.J. Pure Appl. Algebra 193 (2004), 61-70. Zbl 1054.14083, MR 2076378, 10.1016/j.jpaa.2004.03.003 |
| Reference:
|
[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. Zbl 1073.14077, MR 2138860, 10.1090/S0002-9939-05-07570-2 |
| Reference:
|
[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. Zbl 1077.14092, MR 2110519, 10.1016/j.jpaa.2004.08.030 |
| Reference:
|
[8] Liu, D., Sun, X.: Images of higher-order differential operators of polynomial algebras.Bull. Aust. Math. Soc. 96 (2017), 205-211. Zbl 1390.13078, MR 3703902, 10.1017/S0004972717000454 |
| Reference:
|
[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. Zbl 0889.22008, MR 1601155 |
| Reference:
|
[10] Meng, G.: Legendre transform, Hessian conjecture and tree formula.Appl. Math. Lett. 19 (2006), 503-510. Zbl 1132.14340, MR 2221506, 10.1016/j.aml.2005.07.006 |
| Reference:
|
[11] Sun, X.: Images of derivations of polynomial algebras with divergence zero.J. Algebra 492 (2017), 414-418. Zbl 1386.14208, MR 3709158, 10.1016/j.jalgebra.2017.09.020 |
| Reference:
|
[12] Tsuchimoto, Y.: Endomorphisms of Weyl algebra and $p$-curvatures.Osaka J. Math. 42 (2005), 435-452. Zbl 1105.16024, MR 2147727 |
| Reference:
|
[13] Essen, A. van den: Polynomial Automorphisms and the Jacobian Conjecture.Progress in Mathematics 190, Birkhäuser, Basel (2000). Zbl 0962.14037, MR 1790619, 10.1007/978-3-0348-8440-2 |
| Reference:
|
[14] Essen, A. van den, Sun, X.: Monomial preserving derivations and Mathieu-Zhao subspaces.J. Pure Appl. Algebra 222 (2018), 3219-3223. Zbl 06867612, MR 3795641, 10.1016/j.jpaa.2017.12.003 |
| Reference:
|
[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. Zbl 1317.33007, MR 3335985, 10.1016/j.jpaa.2014.12.024 |
| Reference:
|
[16] Essen, A. van den, Wright, D., Zhao, W.: On the image conjecture.J. Algebra 340 (2011), 211-224. Zbl 1235.14057, MR 2813570, 10.1016/j.jalgebra.2011.04.036 |
| Reference:
|
[17] Essen, A. van den, Zhao, W.: Two results on homogeneous Hessian nilpotent polynomials.J. Pure Appl. Algebra 212 (2008), 2190-2193. Zbl 1147.14033, MR 2418165, 10.1016/j.jpaa.2008.01.005 |
| Reference:
|
[18] Wright, D.: The Jacobian conjecture as a problem in combinatorics.Affine Algebraic Geometry Osaka University Press, Osaka (2007), 483-503. Zbl 1129.14087, MR 2330486 |
| Reference:
|
[19] Zhao, W.: A vanishing conjecture on differential operators with constant coefficients.Acta Math. Vietnam. 32 (2007), 259-286. Zbl 1139.14303, MR 2368014 |
| Reference:
|
[20] Zhao, W.: Hessian nilpotent polynomials and the Jacobian conjecture.Trans. Am. Math. Soc. 359 (2007), 249-274. Zbl 1109.14041, MR 2247890, 10.1090/S0002-9947-06-03898-0 |
| Reference:
|
[21] Zhao, W.: Images of commuting differential operators of order one with constant leading coefficients.J. Algebra 324 (2010), 231-247. Zbl 1197.14064, MR 2651354, 10.1016/j.jalgebra.2010.04.022 |
| . |
