Finite flag-transitive affine planes with a solvable automorphism group

TL;DR

This paper classifies finite flag-transitive affine planes with solvable automorphism groups, linking their structure to planar functions and permutation polynomials, and providing new characterizations in odd and even order cases.

Contribution

It introduces a novel approach connecting affine planes with planar functions and permutation polynomials, leading to classifications in specific cases.

Findings

Characterization of the Kantor-Suetake family using generalized twisted fields.

Development of techniques for permutation polynomials of DO type.

Classification of planes with dimensions up to four over their kernels.

Abstract

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, we characterize the Kantor-Suetake family by using Menichetti's classification of generalized twisted fields and Blokhuis, Lavrauw and Ball's classifcation of rank two commutative semifields. In the even order case, we develop a technique to study permutation polynomials of DO type by quadratic forms and characterize such planes that have dimensions up to four over their kernels.