# On permutations induced by tame automorphisms over finite fields

## TL;DR

This paper provides formulas to determine the sign of permutations induced by tame automorphisms over finite fields, enabling easier analysis of their permutation properties based on their decompositions.

## Contribution

It introduces explicit formulas for the permutation sign induced by elementary and affine automorphisms over finite fields, and combines these to analyze tame automorphisms.

## Key findings

- Formulas for signs of elementary automorphisms
- Formulas for signs of affine automorphisms
- Method to determine signs of tame automorphisms

## 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.