A uniqueness problem in simple transcendental extensions of valued fields

Abstract

Let v₀ be a valuation of a field K₀ with value group G₀ and v be an extension of u0 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 teKJ,x)\K₀ with v(t) non-torsion mod Go such that i> is the unique extension to K₀(x) of its restriction to the subfield X₀(t). It is proved that the answer to this question is "yes" if v₀ is henselian or if v₀ is of rank 1 with G₀ a cofinal subset of the value group of v 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, v)/(K₀, v₀).