Eulerian operators and the Jacobian conjecture
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- by Arno van den Essen PDF
- Proc. Amer. Math. Soc. 118 (1993), 373-378 Request permission
Abstract:
In this paper we introduce a new class of polynomial maps, the so-called nice polynomial maps. Using Eulerian operators we show how for these polynomial maps the main results obtained by Bass (Differential structure of étale extensions of polynomial algebras, Proc. Workshop on Commutative Algebra, MSRI, 1987) can be proved in a very simple and elementary way. Furthermore we show that every polynomial map $F$ satisfying the Jacobian condition, det $JF \in {k^{\ast }}$, is equivalent to a nice polynomial map; more precisely the polynomial map ${F_{(\lambda )}}(X) = F(X + \lambda ) - F(\lambda )$ is nice for almost all $\lambda \in {k^n}$.References
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- Kossivi Adjamagbo and Arno van den Essen, Eulerian systems of partial differential equations and the Jacobian conjecture, J. Pure Appl. Algebra 74 (1991), no. 1, 1–15. MR 1129126, DOI 10.1016/0022-4049(91)90045-4 H. Bass, Differential structure of étale extensions of polynomial algebras, Proc. Workshop on Commutative Algebra, MSRI, 1987. A. van den Essen, $\mathcal {D}$-modules and the Jacobian Conjecture, Proc. Internat. Conf. "$\mathcal {D}$-Modules and Microlocal Geometry", Lisbon, Portugal, October 1990 (to appear).
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 373-378
- MSC: Primary 14E07; Secondary 13B10, 13H10, 14E20, 16S30, 16S32
- DOI: https://doi.org/10.1090/S0002-9939-1993-1129883-3
- MathSciNet review: 1129883