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A classification of unitals in nearfield planes with maximal automorphism group

Randon J. Weaver, Robert S. Coulter, Alice M. W. Hui

TL;DR

The paper tackles the problem of classifying parabolic unitals in regular nearfield planes of order $q^2$ whose linear collineation group attains the maximal size $q^3-q$. It develops a framework based on central and linear collineations, derives strong structural restrictions for unitals under the maximal-order assumption, and expresses potential unitals in two canonical families $\mathcal{U}(j)$ and $\mathcal{V}(j)$. Using a combination of geometric configuration arguments (notably against O'Nan configurations) and number-theoretic tools (notably properties of the all-ones polynomial), the authors show that only the Wantz unitals ($j=1$) survive in the maximal-case classification. Consequently, every unitals with maximal linear automorphism group in these regular nearfield planes is projectively equivalent to a Wantz unital, extending known results from Desarguesian and BM-type unitals to regular nearfield planes. The work also supplies broader results about parabolic unitals in nearfield planes under weaker hypotheses and clarifies the structure of linear collineations fixing unitals in this setting.

Abstract

We classify the parabolic unitals in regular nearfield planes of odd order $q^2$ whose linear collineation group has the maximal size of $q^3-q$. We also establish a number of more general results concerning parabolic unitals in regular nearfield planes under weaker assumptions.

A classification of unitals in nearfield planes with maximal automorphism group

TL;DR

The paper tackles the problem of classifying parabolic unitals in regular nearfield planes of order whose linear collineation group attains the maximal size . It develops a framework based on central and linear collineations, derives strong structural restrictions for unitals under the maximal-order assumption, and expresses potential unitals in two canonical families and . Using a combination of geometric configuration arguments (notably against O'Nan configurations) and number-theoretic tools (notably properties of the all-ones polynomial), the authors show that only the Wantz unitals () survive in the maximal-case classification. Consequently, every unitals with maximal linear automorphism group in these regular nearfield planes is projectively equivalent to a Wantz unital, extending known results from Desarguesian and BM-type unitals to regular nearfield planes. The work also supplies broader results about parabolic unitals in nearfield planes under weaker hypotheses and clarifies the structure of linear collineations fixing unitals in this setting.

Abstract

We classify the parabolic unitals in regular nearfield planes of odd order whose linear collineation group has the maximal size of . We also establish a number of more general results concerning parabolic unitals in regular nearfield planes under weaker assumptions.

Paper Structure

This paper contains 9 sections, 25 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Nearfields
  3. Nearfield planes and their collineations
  4. Central collineations that fix a unital
  5. Restrictions on the linear collineations of unitals in $\textgoth{N}$
  6. The non-existence of certain O'Nan configurations in unitals in $\textgoth{N}$
  7. The sets $B(a,b)$ and the lines of $\textgoth{N}$
  8. The all-ones polynomial over ${\mathbb F}_{q}$ -- an interlude
  9. Unitals containing $(0,1,0)$ in $\textgoth{N}$ with the largest possible linear collineation group

Key Result

Lemma 2

Let $\mathds{N}=\mathds{N}(2,q)$ and let $\mathfrak{g}$ be a primitive element of ${\mathbb F}_{q^2}$. If $S<\langle \mathds{N}^\star,\star \rangle$, then either

Theorems & Definitions (47)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5: A-1955-peuf, Satz 12
  • Lemma 6: A-1955-peuf, Satz 13
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 37 more