The problem of constructing, analyzing and classifying extensions of a valuation from a base field to its rational function field has generated an amazing variety of approaches and results. However, there are still open problems and new approaches. I will discuss the special case of a tame base field, which allows us to prove particularly strong results. Also the case of tame algebraic, not necessarily simple, extensions of an arbitrary henselian base field will be treated. The idea is to first construct suitable extensions of the base field by homogenous elements, introduced in 2004 in the paper "Value groups, residue fields and bad places of rational function fields", Trans. Amer. Math. Soc. 356, and then obtain key polynomials as their minimal polynomials.
This is joint work with Arpan Dutta.