Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Inverse degree of an affine space triangular automorphism


Author: Shu Kawaguchi
Journal: Proc. Amer. Math. Soc. 141 (2013), 3353-3360
MSC (2010): Primary 08A35, 13B25, 14J50, 14R10
DOI: https://doi.org/10.1090/S0002-9939-2013-11631-X
Published electronically: June 17, 2013
MathSciNet review: 3080158
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any $ \mathbb{Q}$-algebra $ R$ and any triangular automorphism $ f: R^N\to R^N$ with Jacobian one on the affine space, we show that $ \deg (f^{-1})$ is bounded from above by a constant depending only on $ N$ and $ \deg (f)$. This is seen as a generalization of a result by Furter on the affine plane. Our proof uses (a version of) Furter's estimate on nilpotency indices and Abhyankar-Gurjar's formal inversion formula. It follows that when the Jacobian of a triangular automorphism $ f: R^N\to R^N$ is not necessarily equal to one, $ \deg (f^{-1})$ is bounded from above by a constant depending only on $ N$, $ \deg (f)$ and $ \deg (1/\operatorname {Jac}(f))$.


References [Enhancements On Off] (What's this?)

  • 1. S. S. Abhyankar, Lectures in algebraic geometry. Notes by Chris Christensen, Purdue University, 1974.
  • 2. H. Bass, The Jacobian conjecture and inverse degrees, Arithmetic and geometry, Vol. II, 65-75, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983. MR 717606 (84k:13007)
  • 3. H. Bass, E. H. Connel, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287-330. MR 663785 (83k:14028)
  • 4. C. C.-A. Cheng, S. S.-S. Wang, J.-T. Yu, Degree bounds for inverses of polynomial automorphisms, Proc. Amer. Math. Soc. 120 (1994), no. 3, 705-707. MR 1195715 (94e:14016)
  • 5. H. Derksen, Inverse degrees and the Jacobian conjecture, Comm. Algebra 22 (1994), no. 12, 4793-4794. MR 1285708 (95b:14011)
  • 6. A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. MR 1790619 (2001j:14082)
  • 7. M. Fouenié, J.-Ph. Furter and D. Pinchon, Computation of the maximal degree of the inverse of a cubic automorphism of the affine plane with Jacobian $ 1$ via Gröbner bases, J. Symbolic Comput. 26 (1998), no. 3, 381-386. MR 1633884 (2000b:14085)
  • 8. J.-Ph. Furter, On the degree of the inverse of an automorphism of the affine plane, J. Pure Appl. Algebra 130 (1998), no. 3, 277-292. MR 1633771 (99g:14015)
  • 9. S. Maubach, The automorphism group of $ \mathbb{C} [T]/(T^m)[X_1,\dots ,X_n]$, Comm. Algebra 30 (2002), no. 2, 619-629. MR 1883016 (2002k:14092)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 08A35, 13B25, 14J50, 14R10

Retrieve articles in all journals with MSC (2010): 08A35, 13B25, 14J50, 14R10


Additional Information

Shu Kawaguchi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email: kawaguch@math.sci.osaka-u.ac.jp, kawaguch@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2013-11631-X
Keywords: Automorphism, degree, inverse, polynomial ring, non-reduced ring
Received by editor(s): October 18, 2010
Received by editor(s) in revised form: December 13, 2011
Published electronically: June 17, 2013
Additional Notes: This work is partially supported by KAKENHI 21740018
Communicated by: Harm Derksen
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society