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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.05431 |
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| _version_ | 1866917127534936064 |
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| author | Gemmell, Zoë Trudgian, Tim |
| author_facet | Gemmell, Zoë Trudgian, Tim |
| contents | A function from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ is $k$th order sum-free if the sum of its values over each $k$-dimensional $\mathbb{F}_2$-affine subspace is nonzero. It is conjectured that for $n$ odd and prime, $f_\textrm{inv}=x^{-1}$ is not $k$th order sum-free for $3 \leq k \leq n-3$. This is the unresolved part of Carlet's conjecture, which gives exact values for which $f_\textrm{inv}$ is $k$th order sum-free. We give two results as improvements on an explicit estimate on the number of $q$-rational points of an $\mathbb{F}_q$-definable hypersurface previously proved by Cafure and Matera. We use these results to prove that $f_\textrm{inv}$ is not $k$th order sum-free for $3\leq k \leq \frac{3}{13}n+0.461$, improving on work previously done by Hou and Zhao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Applying hypersurface bounds to a conjecture by Carlet Gemmell, Zoë Trudgian, Tim Number Theory A function from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ is $k$th order sum-free if the sum of its values over each $k$-dimensional $\mathbb{F}_2$-affine subspace is nonzero. It is conjectured that for $n$ odd and prime, $f_\textrm{inv}=x^{-1}$ is not $k$th order sum-free for $3 \leq k \leq n-3$. This is the unresolved part of Carlet's conjecture, which gives exact values for which $f_\textrm{inv}$ is $k$th order sum-free. We give two results as improvements on an explicit estimate on the number of $q$-rational points of an $\mathbb{F}_q$-definable hypersurface previously proved by Cafure and Matera. We use these results to prove that $f_\textrm{inv}$ is not $k$th order sum-free for $3\leq k \leq \frac{3}{13}n+0.461$, improving on work previously done by Hou and Zhao. |
| title | Applying hypersurface bounds to a conjecture by Carlet |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.05431 |