Dimension of definable sets, algebraic boundedness and Henselian fields (Q916646)

For an integral domain \(D\), the algebraic dimension \(\operatorname{algdim}(S)\) of an arbitrary subset \(S\subseteq D^m\) is defined as the maximum number of polynomial functions on \(S\) that can be algebraically independent over \(D\). The author shows that if an expansion of \(D\) satisfies a certain simple algebraic condition (called algebraic boundedness) then the algebraic dimension defines a dimension function on its Tarski system of definable sets. The main result of this paper is the following: Each Henselian field of characteristic 0 is algebraically bounded and the algebraic dimension is the only dimension function on its Tarski system of definable sets. (There is no restriction on the characteristic of the residue field.)