Minimal limit key polynomials

In this paper, we extend the theory of minimal limit key polynomials of valuations on the polynomial ring $\text{kx}$. We use the theory of cuts on ordered abelian groups to show that the previous results on bounded sets of key polynomials of rank-one valuations, extend to vertically bounded sets of key polynomials of valuations of an arbitrary rank. We discuss as well properties of minimal limit key polynomials in the vertically unbounded case.