Inversion and subspaces of a finite field |
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Authors: | Email author" target="_blank">Sandro?MattareiEmail author |
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Institution: | 1.Università degli Studi di Trento,Povo (Trento),Italy |
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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. |
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