New necessary conditions for the existence of finite non-Desarguesian flag-transitive projective plane

TL;DR

This paper establishes necessary polynomial conditions over finite fields of characteristic 3 for the existence of finite non-Desarguesian flag-transitive projective planes, contributing to the conjecture that all such planes are Desarguesian.

Contribution

It introduces new algebraic necessary conditions that must be satisfied for non-Desarguesian flag-transitive projective planes to exist, advancing understanding of their structure.

Findings

Provides polynomial equations over finite fields of characteristic 3

Identifies conditions that exclude the existence of certain non-Desarguesian planes

Supports the conjecture that all flag-transitive projective planes are Desarguesian

Abstract

This paper studies the existence of finite non-Desarguesian flag-transitive projective plane, giving necessary conditions in terms of polynomial equations over finite fields of characteristic $3$. This sheds light on the longstanding conjecture that every finite flag-transitive projective plane is Desarguesian.