{"id":"hexagon/2611.00032","pdf":"https://hexagonmath.org/pdf/2611.00032v1","url":"https://hexagonmath.org/abs/2611.00032v1","tags":["cs.FL","msc:11B85","msc:11J70","msc:11C20","msc:68R15","Hankel determinants","automatic sequences","continued fractions","power series over finite fields"],"score":0,"title":"The Hankel determinants of an automatic sequence need not be automatic","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a $d$-automatic sequence must be $d$-regular, or must become $d$-automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo $3$. Let $\\alpha$ be the root of $f^4+f^2-Tf+1=0$ in $\\mathbb{F}_3((1/T))$ with $|\\alpha|<1$, the quartic of Mills and Robbins, and let $S$ be its coefficient sequence, which is $3$-automatic by Christol's theorem. The Hankel determinants of $S$ vanish in runs whose lengths are unbounded but of the order of the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}