Inverse degree of an affine space triangular automorphism
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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
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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
- MR Author ID: 655244
- Email: kawaguch@math.sci.osaka-u.ac.jp, kawaguch@math.kyoto-u.ac.jp
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3080158