manuscripta math. 65, 357 - 376 (1989) manuscripta mathematica @ Sprmger-Verlag 1989 On valued function fields I B. Green, M. Matignon and F. Pop 1 The purpose of this work is to study valued function fields of one variable over a constant field, that is function fields of one variable which are equipped with an arbitrary valuation on the constant field and a prolongation to the function field, such that the corresponding residue fields again are a function field in one variable. The first part of the work, that is this paper, deals with the defect and genus inequality of a valued function field. In a second paper we shall define a divisor reduction map and show the existence of elements with the uniqueness property over an arbitrary henselian base field. By an element with the uniqueness property for a valued function field we mean an element such that the valuation on the function field is the unique prolongation of the functional (Gaussian) valuation on the rational function field obtained by adjoining this element to the constant field. To this paper: In the first section we recall briefly those definitions and gen- era] facts needed from the (Deuring's) reduction theory of valued function fields and prove a valuation decomposition lemma which will be fundamental for latter sections of the paper. In the second section we define and study the vector space defect of a non- archimedean valued vector space over a valued field. We shall show that for valued function fields this defect represents a geometric property for the reduction and that over a henselian constant field it coincides with the henselian defect for the valued function field. Using the results of this section a simple direct proof that for valued function fields of genus greater than 1, the genus and residual genus being equal is sufficient to ensure good reduction, is obtained. We also study the reduction of a valued function field when the genus and residual genus are one and obtain sufficient conditions for ensuring good reduction in this case. z The first author gratefully acknowledges support from the University of Stellenbosch, South Africa 357