On Permutations Induced by Tame Automorphisms Over Finite Fields

Abstract

The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K16066.

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Correspondence to Keisuke Hakuta.

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Hakuta, K. On Permutations Induced by Tame Automorphisms Over Finite Fields. Acta Math Vietnam 43, 309–324 (2018). https://doi.org/10.1007/s40306-017-0217-0

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Keywords

  • Affine algebraic geometry
  • Polynomial automorphism
  • Tame automorphism
  • Finite field
  • Permutation

Mathematics Subject Classification (2010)

  • 14R10
  • 12E20
  • 20B25