Abstract
We show that the GVC (generalized vanishing conjecture) holds for the differential operator Λ = (∂x − Φ(∂y))∂y and all polynomials P(x,y), where Φ(t) is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
A. Belov-Kanel, M. Kontsevich: The Jacobian conjecture is stably equivalent to the Dixmier conjecture. Mosc. Math. J. 7 (2007), 209–218.
M. de Bondt: A few remarks on the generalized vanishing conjecture. Arch. Math. 100 (2013), 533–538.
M. de Bondt, A. van den Essen: Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. J. Pure Appl. Algebra 193 (2004), 61–70.
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.
M. de Bondt, A. van den Essen: Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. II. J. Pure Appl. Algebra 196 (2005), 135–148.
D. Liu, X. Sun: Images of higher-order differential operators of polynomial algebras. Bull. Aust. Math. Soc. 96 (2017), 205–211.
O. Mathieu: Some conjectures about invariant theory and their applications. Algebre non commutative, groupes quantiques et invariants (Alev, J. et al., eds.). Semin. Congr. 2, Societe Mathematique de France, Paris, 1995, pp. 263–279.
G. Meng: Legendre transform, Hessian conjecture and tree formula. Appl. Math. Lett. 19 (2006), 503–510.
X. Sun: Images of derivations of polynomial algebras with divergence zero. J. Algebra 492 (2017), 414–418.
Y. Tsuchimoto: Endomorphisms of Weyl algebra and p-curvatures. Osaka J. Math. 42 (2005), 435–452.
A. van den Essen: Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics 190, Birkhauser, Basel, 2000.
A. van den Essen, X. Sun: Monomial preserving derivations and Mathieu-Zhao subspaces. J. Pure Appl. Algebra 222 (2018), 3219–3223.
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.
A. van den Essen, D. Wright, W. Zhao: On the image conjecture. J. Algebra 340 (2011), 211–224.
A. van den Essen, W. Zhao: Two results on homogeneous Hessian nilpotent polynomials. J. Pure Appl. Algebra 212 (2008), 2190–2193.
D. Wright: The Jacobian conjecture as a problem in combinatorics. Affine Algebraic Geometry. Osaka University Press, Osaka, 2007, pp. 483–503.
W. Zhao: A vanishing conjecture on differential operators with constant coefficients. Acta Math. Vietnam. 32 (2007), 259–286.
W. Zhao: Hessian nilpotent polynomials and the Jacobian conjecture. Trans. Am. Math. Soc. 359 (2007), 249–274.
W. Zhao: Images of commuting differential operators of order one with constant leading coefficients. J. Algebra 324 (2010), 231–247.
Acknowledgement
The paper is a part of the first author’s Master’s Thesis under the supervision of the second author.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author has been partially supported by the NSF of China (11771176, 11871241).
Rights and permissions
About this article
Cite this article
Feng, Z., Sun, X. On the generalized vanishing conjecture. Czech Math J 69, 1061–1068 (2019). https://doi.org/10.21136/CMJ.2019.0049-18
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2019.0049-18