{"id":"hexagon/2610.00533","pdf":"https://hexagonmath.org/pdf/2610.00533v1","url":"https://hexagonmath.org/abs/2610.00533v1","tags":["math.AG","msc:51E21","Plane arcs"],"score":0,"title":"Complete planar arcs over nonsquare binary fields","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Jose Voloch"],"reviews":0,"subject":"math.CO","summary":"Let $m'_2(q)$ be the largest size of a complete arc in $\\operatorname{PG}(2,q)$ 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,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}