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arxiv: 2606.11763 · v1 · pith:GRJJDXCE · submitted 2026-06-10 · math.CO

Enumerating inherited conics in Andr\'e planes of odd order

S.G. Barwick , Alice M.W. Hui , Wen-Ai Jackson This is my paper

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classification math.CO
keywords André planesinherited conicsarcsfinite projective planesodd orderprime exponentDesarguesian planes
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The pith

Conics in PG(2,q^t) inherit to arcs in the André plane precisely when their intersection with the replaced André net meets a controlled pattern, for odd q and prime t.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the known inheritance result for Hall planes to the general case of André planes obtained by replacing an André net in the Desarguesian plane PG(2,q^t) when t is prime and q is odd. It determines the exact intersection conditions under which a conic remains an arc after replacement. The authors then give an explicit enumeration of how many such inherited arcs exist in the resulting André plane. A sympathetic reader cares because the result supplies a concrete, countable supply of arcs in these non-Desarguesian planes that descend directly from familiar conics.

Core claim

When q is odd and t is prime, a conic in PG(2,q^t) inherits to an arc in the André plane obtained by replacing an André net if and only if it intersects that net in a manner that preserves the arc property after replacement. The total number of such inherited arcs is then enumerated explicitly in terms of q and t.

What carries the argument

The inheritance condition determined by the intersection pattern of the conic with the replaced André net.

If this is right

  • The enumeration supplies an exact count of inherited arcs for every André plane of this type.
  • The result extends the Hall-plane inheritance theorem uniformly to all prime exponents t greater than or equal to 3.
  • Only conics whose intersections with the André net fall into the allowed configurations contribute to the count.
  • The same inheritance criterion applies across all choices of André net in PG(2,q^t).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The enumerated arcs could serve as explicit examples when studying ovals or blocking sets inside André planes.
  • The intersection criterion might be tested computationally for small q and t to confirm the formulas.
  • Analogous inheritance questions remain open for even q or composite t.

Load-bearing premise

The inheritance condition for a conic depends only on its intersection pattern with the replaced André net in a manner that extends directly from the t=2 Hall plane case when t is prime.

What would settle it

For q=3 and t=3, list all conics in PG(2,27), determine which inherit under the paper's intersection rule, and check whether their count matches the enumerated formula.

read the original abstract

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an Andr\'e net of $PG(2,q^t)$, $t\geq 3$ to construct an Andr\'e plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an Andr\'e plane. Further, the number of arcs in an Andr\'e plane that are inherited in this way is enumerated.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the conditions under which a conic in the Desarguesian plane PG(2,q^t) inherits to an arc in an André plane of order q^t (q odd, t prime) obtained by replacing an André net, via analysis of intersection patterns with the net; it further enumerates the resulting inherited arcs, extending the known t=2 (Hall plane) case.

Significance. If the determination and enumeration hold, the work supplies an explicit extension of the t=2 solution to all prime t≥3, yielding concrete counts of inherited arcs that may support further classification results in finite geometry. The restriction to prime t enables the required algebraic simplifications, which is a methodological strength.

minor comments (3)
  1. [Abstract] The abstract states that the number of inherited arcs is enumerated but does not record the explicit formula; including the closed-form count (in terms of q and t) would improve immediate readability.
  2. Notation for the André net and the replacement map is introduced without a dedicated preliminary subsection; a short table or diagram contrasting the t=2 and general-prime cases would aid readers.
  3. Several intersection lemmas are stated for conics meeting the net in specific patterns; cross-references to the corresponding statements in the t=2 literature would clarify the extension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, including the assessment of its significance in extending the t=2 case to prime t, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper determines inheritance conditions for conics in André planes of order q^t (q odd, t prime) by analyzing intersection patterns with the replaced André net, generalizing the t=2 Hall plane case via direct geometric extension. The enumeration of resulting arcs follows from these intersection conditions and algebraic simplifications enabled by primeness of t. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central claims rest on independent geometric and combinatorial arguments that are externally verifiable against the Desarguesian plane structure. The derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of finite fields and projective planes together with the definition of André net replacement; no new entities or fitted parameters are indicated in the abstract.

axioms (2)
  • standard math Algebraic properties of finite fields of odd characteristic and their projective planes PG(2,q^t)
    Required to define conics and the ambient Desarguesian plane.
  • domain assumption Existence and replacement rules for André nets in PG(2,q^t) when t is prime
    Central to the construction of the André plane and the inheritance analysis.

pith-pipeline@v0.9.1-grok · 5672 in / 1348 out tokens · 31733 ms · 2026-06-27T09:26:48.151597+00:00 · methodology

discussion (0)

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Reference graph

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