Residual Transcendental Extensions of A

Let K be a field and v a valuation on K. We discuss some properties of extensions w of v to the field K(X1, . . . , Xn) of rational functions in n variables over K, for which the residue field of w has transcendence degree n over the residue field of v. 2000 Mathematics Subject Classification: 12J20, 12F20. 1.Introduction LetK be a field and v a valuation onK. Residual transcendental extensions of v to the field K(X) of rational functions in one variable over K have been studied in a number of articles, including [8], [9], [10], [11], [13], [1], [2], [3] and [14]. Nagata [8] conjectured that if w is a residual transcendental extension of v to K(X), then the residue field kw of w is a simple transcendental extension of a finite algebraic extension of the residue field kv of v. The problem was solved by Ohm [10] and by Popescu [13]. Further questions on residual transcendental extensions of a valuation have been considered by Ohm, and in the process he stated in [11] three conjectures concerning some natural numbers like ramification index and residual degree. These three problems were later solved in [1]. The main ingredient used in [1] to investigate these and other related questions was a theorem of characterization of residual transcendental extensions of v toK(X). The situation is a lot more complicated if one replaces K(X) by the field K(X1, . . . , Xn) of rational functions in n variables over K and one attempts to describe all the extensions w of v to K(X1, . . . , Xn) for which the residue field kw of w has transcendence degree n over the residue field kv of v. One could of course use the characterization theorem from [1] mentioned above in order to describe the valuation w on K(X1, . . . , Xn) in n steps, by taking a residual transcendental extension of v to K(X1), followed