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The Welch function is not third-order sum-free

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorycs.IT — Information Theory

Contributed by Deep Bhattacharjee

Version 2 / Oct 22, 2026 / CC BY 4.0

Abstract

Let n=2m+1n=2m+1 be odd and let $\F_{2^n}$ be the field with 2n2^n elements. A function on $\F_{2^n}$ is third-order sum-free when its values over any eight elements forming a three-dimensional affine subspace never add up to zero. The Welch function is the power map x↦x2m+3x\mapsto x^{2^m+3}; it is one of the classical almost perfect nonlinear functions, and Hou and Zhao conjectured that for n≥7n\ge7 it is not third-order sum-free. We prove this. The question comes down to a linearised polynomial g(x,y)g(x,y) in yy whose roots form a vector space Z(x)Z(x) over $\F_2$ containing 11 and xx, and the Welch function fails to be third-order sum-free exactly when Z(x)Z(x) has dimension three for some xx outside $\F_2$. We show that the dimension is always 22 or 33, and that it is 33 if and only if two explicit trace conditions hold: with u=x2mu=x^{2^m}, both $\Tr(1/(x+u))$ and $\Tr(xu/(x+u))$ equal 11. The proof applies the field automorphism z↦z2m+1z\mapsto z^{2^{m+1}}, whose square is the Frobenius map, to turn gg into an additive polynomial of degree eight whose roots always lie in $\F_{2^n}$. Counting the elements that satisfy both trace conditions gives exactly (2n+1−3Kn)/4(2^n+1-3K_n)/4 of them, where KnK_n is the binary Kloosterman sum of $\F_{2^n}$, and Weil's bound ∣Kn∣≤2n/2+1|K_n|\le 2^{n/2+1} makes this count positive for every n≥7n\ge7. The Welch function sums to zero over exactly (2n+1−3Kn)(2n−1)/168(2^n+1-3K_n)(2^n-1)/168 three-dimensional linear subspaces, and the count 2+(2n+1−3Kn)/42+(2^n+1-3K_n)/4 agrees with the numbers that Hou and Zhao computed for n≤15n\le15. Every statement is proved by hand.

Provenance statement

Deep Bhattacharjee chose the problem, Conjecture 4.10 of Hou and Zhao (arXiv:2609.31489), and directed the work. Claude (Anthropic), through Claude Code, searched the literature, found and checked the proofs, wrote the verification programs and drafted the paper. The results of this paper were first made public on 9 October 2026, as release \texttt{v1.0.0} of the repository \url{https://github.com/creelie/eConjectureWelchSumFree}; later versions (\texttt{v1.1.0} of 22 October and \texttt{v1.2.0} of 26 October) are archived on Zenodo as \href{https://doi.org/10.5281/zenodo.23553010}{10.5281/zenodo.23553010} and \href{https://doi.org/10.5281/zenodo.23662272}{10.5281/zenodo.23662272} (concept DOI \href{https://doi.org/10.5281/zenodo.23553009}{10.5281/zenodo.23553009}). An earlier and incomplete version of the paper appeared on Hexagon on 22 October 2026 as the preprint \texttt{hexagon:2610.00472}, available at \url{https://hexagonmath.org/2610.00472}. That version cited the conjecture and the corollary of Hou and Zhao as Conjecture~4.10 and Corollary~4.9, numbers that do not match their paper, listed the programs that check its statements in the text, and had a short abstract. This version is complete and self-contained: the references follow the arXiv paper of Hou and Zhao (Conjecture~5.10, Corollary~5.9, Section~5.3), every reference has been checked at its source, the abstract states what is proved, and every statement is proved by hand, so that no statement rests on a computer. The source of this revision is in release \texttt{v2.0.0} of the same repository, archived as \href{https://doi.org/10.5281/zenodo.23663217}{10.5281/zenodo.23663217}, with programs that recompute Table~\ref{tab} and check the criterion on small fields.

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

References

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 22, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 26, 2026