Abstract
For a finite field \(\mathbb{F}_{q}\), the set of polynomial endomorphisms of \(\mathbb{F}_{q}^{n}\) of degree d is bounded when n and d are fixed. This makes it possible to compute the set of all polynomial automorphisms of degree d or less (while it is still an open problem to determine generators of the group of polynomial automorphisms). In this chapter, we do exactly that: we compute the set of all automorphisms for the dimensions and degrees for which it is computationally feasible. In addition, we study a slightly larger class of endomorphisms, the “mock automorphisms,” which are Keller maps inducing bijections of the space \(\mathbb{F}_{q}^{n}\) (essentially characteristic p counterexamples to the Jacobian Conjecture which are injective) and determine some of their equivalence classes. We also determine equivalence classes of locally finite polynomial endomorphisms of low degree. The results of this chapter are mainly of a computational nature, and the conjectures we can make due to these computations, but we also prove a few theoretical results related to mock automorphisms.
AMS classification: 14R10, 14Q15, 37P05
Research partially done while funded by Veni-grant of council for the physical sciences, Netherlands Organisation for scientific research (NWO).
Funded by Phd-grant of council for the physical sciences, Netherlands Organisation for scientific research (NWO).
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Notes
- 1.
Even with many extra factors of computing power, the sets grow in size so fast, that the gain will be minimal.
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Acknowledgements
The authors would like to thank Joost Berson for some useful discussions. Furthermore, we would like to express our gratitude to the two referees, who have significantly improved the readability of the paper on many fronts. Thank you very much!
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Maubach, S., Willems, R. (2014). Keller Maps of Low Degree over Finite Fields. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_26
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DOI: https://doi.org/10.1007/978-3-319-05681-4_26
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