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Counting zero-sum subspaces for the multiplicative inverse function
Abstract
Let q=2n with n≥6, and let 𝔽q be the finite field with q elements. For a subspace E of 𝔽q, define S(E)=∑x∈ E∖{0}x−1. Let Nn,k denote the number of k-dimensional 𝔽2-subspaces E for which S(E)=0. We prove that, if k≥3 and n≥2k+1, then |2nNn,k/genfrac[]0ptnk2−1|<2r2+r2n(1−r/2), where r=⌊(k−1)/2⌋, and genfrac[]0ptnk2 is the Gaussian binomial coefficient. In particular, Nn,k∼2−ngenfrac[]0ptnk2 (n→∞) uniformly for n/3<k≤(n−1)/2. Combined with the known low-dimensional cases and the symmetry between dimensions k and n−k, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every 3≤ k≤ n−3, there exists a k-dimensional 𝔽2-subspace E such that S(E)=0.
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BibTeX
@misc{cheng2026counting,
title = {{Counting zero-sum subspaces for the multiplicative inverse function}},
author = {Kaimin Cheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06445},
eprint = {2610.06445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06445},
}
RIS
TY - UNPB
TI - Counting zero-sum subspaces for the multiplicative inverse function
AU - Kaimin Cheng
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06445
UR - https://arxiv.org/abs/2610.06445
AB - Let q=2^n with n≥6, and let 𝔽_q be the finite field with q elements. For a subspace E of 𝔽_q, define S(E)=∑_x∈ E∖{0}x^−1. Let N_n,k denote the number of k-dimensional 𝔽_2-subspaces E for which S(E)=0. We prove that, if k≥3 and n≥2k+1, then |2^nN_n,k/genfrac[]0ptnk_2−1|<2^r^2+r2^n(1−r/2), where r=⌊(k−1)/2⌋, and genfrac[]0ptnk_2 is the Gaussian binomial coefficient. In particular, N_n,k∼2^−ngenfrac[]0ptnk_2 (n→∞) uniformly for n/3<k≤(n−1)/2. Combined with the known low-dimensional cases and the symmetry between dimensions k and n−k, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every 3≤ k≤ n−3, there exists a k-dimensional 𝔽_2-subspace E such that S(E)=0.
ER -