On Permutations Induced by Tame Automorphisms Over Finite Fields
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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.
KeywordsAffine algebraic geometry Polynomial automorphism Tame automorphism Finite field Permutation
Mathematics Subject Classification (2010)14R10 12E20 20B25
This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K16066.
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