Applying hypersurface bounds to a conjecture by Carlet

2025/12/05 by Gemmell, Zoë, Trudgian, Tim
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.05431

Abstract

A function from \mathbbF_2ⁿ to \mathbbF_2ⁿ is kth order sum-free if the sum of its values over each k-dimensional \mathbbF₂-affine subspace is nonzero. It is conjectured that for n odd and prime, f_\textrminv=x^-1 is not kth order sum-free for 3 ≤ k ≤ n-3. This is the unresolved part of Carlet's conjecture, which gives exact values for which f_\textrminv is kth order sum-free. We give two results as improvements on an explicit estimate on the number of q-rational points of an \mathbbF_q-definable hypersurface previously proved by Cafure and Matera. We use these results to prove that f_\textrminv is not kth order sum-free for 3≤ k ≤ (3)/(13)n+0.461, improving on work previously done by Hou and Zhao.

Citations

• On a Conjecture About the Sum-Freedom of the Binary Multiplicative Inverse Function
• More on the sum-freedom of the multiplicative inverse function
• Improved explicit estimates on the number of solutions of equations over a finite field

Related

• On Sum-Free Functions
• Hypersurfaces achieving the Homma-Kim bound
• On a Conjecture About the Sum-Freedom of the Binary Multiplicative Inverse Function
• More on the sum-freedom of the multiplicative inverse function
• On maximal and minimal hypersurfaces of Fermat type