# Inversion and subspaces of a finite field

Mattarei, Sandro (2015) Inversion and subspaces of a finite field. Israel Journal of Mathematics, 206 (1). pp. 327-351. ISSN 0021-2172

Full content URL: https://doi.org/10.1007/s11856-014-1142-8

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## Abstract

Consider two F q -subspaces A and B of a finite field, of the same size, and let A −1 denote the set of inverses of the nonzero elements of A. The author proved that A −1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A −1 ⊈ B, then the bound |A −1∩B| ≤ 2|B|/q − 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q 3. Our main result is a proof of the stronger bound |A −1 ∩ B| ≤ |B|/q · (1 + O d (q −1/2)), for |B| = q d with d > 3. We also classify all examples with |B| ≤ q 3 which attain equality or near-equality in Csajbók’s bound.

Keywords: finite field, Subspace, Inversion, JCNotOpen G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics Publisher 18498 09 Sep 2015 14:49

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