HEXAGONbeta

Complete planar arcs over nonsquare binary fields

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.AG — Algebraic Geometry

Contributed by Jose Voloch ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Let m2′(q)m'_2(q) be the largest size of a complete arc in PG⁡(2,q)\operatorname{PG}(2,q) other than a hyperoval. We prove $m'_2(q) \le q-\floor{q^{2/3}-q^{1/3}}, q=2^h\ge512,\ h\text{ odd}$. The geometric statement underlying the proof is a degree obstruction for arbitrary geometrically irreducible Frobenius nonclassical plane curves; an origin from an arc is unnecessary. The argument uses the curve and a Frobenius twist of its dual, exact weighted point counts, and a local conductor inequality that allows arbitrary singularities. We give a sharper bound obtained by optimizing over powers of two, determine its leading constant according to h mod 3h\bmod3, and prove further refinements using a characteristic-two normal form. Exact finite computations give, for example, m2′(128)≤105m'_2(128)\le105, m2′(512)≤452m'_2(512)\le452, and m2′(2048)≤1875m'_2(2048)\le1875.

Provenance statement

This paper is AI generated. I had some notes making the modest improvement of my old bound for the size of a complete arc, not an hyperoval, for $q$ even and nonsquare, following the same strategy as the old result, which I uploaded to ChatGPT 5.6 to check the proof and suggest possible improvements. After some back and forth, ChatGPT generated the argument below using the dual curve, substantially improving the bound. I also had Claude Opus 5 run an audit of the manuscript and included corrections from the output of this audit. I read through the manuscript and verified several, but not, all arguments.

Tools used

OpenAI
ChatGPT
Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by Jose VolochInitial depositCurrentOct 24, 2026