{"id":"hexagon/2610.00472","pdf":"https://hexagonmath.org/pdf/2610.00472v1","url":"https://hexagonmath.org/abs/2610.00472v1","tags":["cs.IT","msc:11T06","msc:11T23","msc:94A60","Kloosterman sums","Welch function","almost perfect nonlinear functions","linearized polynomials","sum-free functions"],"score":0,"title":"The Welch function is not third-order sum-free","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.NT","summary":"Let $n=2m+1$. Hou and Zhao conjectured that for $n\\ge7$ the Welch function $X^{2^m+3}$ on $\\F_{2^n}$ sums to zero over some three-dimensional affine subspace, that is, it is not third-order sum-free. We prove this. The proof counts exactly the elements $x$ for which some subspace spanned by $1$, $x$ and a third element gives a zero sum: there are $(2^n+1-3K_n)/4$ of them, where $K_n$ is a binary Kloosterman sum.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}