On the product of vector spaces in a commutative field extension

Let K ⊂ L be a commutative field extension. Given K-subspaces A , B of L, we consider the subspace 〈 A B 〉 spanned by the product set A B = { a b | a ∈ A , b ∈ B } . If dim K A = r and dim K B = s , how small can the dimension of 〈 A B 〉 be? In this paper we give a complete answer to this question i...

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Bibliographic Details
Published in:Journal of number theory 2009-02, Vol.129 (2), p.339-348
Main Authors: Eliahou, Shalom, Kervaire, Michel, Lecouvey, Cédric
Format: Article
Language:English
Subjects:
Additive number theory
Combinatorics
Commutative field extension
Mathematics
Number Theory
Product set
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Summary:Let K ⊂ L be a commutative field extension. Given K-subspaces A , B of L, we consider the subspace 〈 A B 〉 spanned by the product set A B = { a b | a ∈ A , b ∈ B } . If dim K A = r and dim K B = s , how small can the dimension of 〈 A B 〉 be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on dim K 〈 A B 〉 turns out, in this case, to be provided by the numerical function κ K , L ( r , s ) = min h ( ⌈ r / h ⌉ + ⌈ s / h ⌉ − 1 ) h , where h runs over the set of K-dimensions of all finite-dimensional intermediate fields K ⊂ H ⊂ L . This bound is closely related to one appearing in additive number theory.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2008.06.004