# On Permutations Induced by Tame Automorphisms Over Finite Fields

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## 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.

## Keywords

Affine algebraic geometry Polynomial automorphism Tame automorphism Finite field Permutation## Mathematics Subject Classification (2010)

14R10 12E20 20B25## Notes

### Acknowledgements

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

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