On the product of vector spaces in a commutative field extension

Shalom Eliahou (1) , Michel Kervaire (2) , Cédric Lecouvey (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 Section de mathématiques [Genève]

Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $\langle AB \rangle$ spanned by the product set $AB=\{ab \mid a \in A, b \in B\}$. If $\dim_K A = r$ and $\dim_K B = s$, how small can the dimension of $\langle AB \rangle$ 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 $\dim_K \langle AB \rangle$ turns out, in this case, to be provided by the numerical function $$ \kappa_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h, $$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \subset H \subset L$. This bound is closely related to one appearing in additive number theory.

Mots clés: product set; vector space; field extension; additive number theory; optimal lower bound

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Shalom Eliahou, Michel Kervaire, Cédric Lecouvey. On the product of vector spaces in a commutative field extension. Journal of Number Theory, 2009, 129 (2), pp.339-348. ⟨10.1016/j.jnt.2008.06.004⟩. ⟨hal-00259373⟩