A UNIQUENESS PROBLEM IN SIMPLE TRANSCENDENTAL EXTENSIONS OF VALUED FIELDS

Let upsilon0 be a valuation of a field K0 with value group G0 and upsilon be an extension of upsilon0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t is-an-element-of K0(x)\K0 with upsilon(t) non-torsion mod G0 such that upsilon is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is ''yes'' if upsilon0 is henselian or if upsilon0 is of rank 1 with G0 a cofinal subset of the value group of upsilon in the latter case, and that it is ''no'' in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, upsilon)/(K0, upsilon0).