Mattarei, Sandro
(2015)
Inversion and subspaces of a finite field.
Israel Journal of Mathematics, 206
(1).
pp. 327351.
ISSN 00212172
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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 ≤ 2B/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 nearequality in Csajbók’s bound.
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