How to determine the sign of valuation on \(\mathbb{C}[x,y]\) (Q2363549)

A divisorial semidegree \(\delta\) on the fraction field \(\mathbb{C}(x,y)\) is the negative of a divisorial valuation \(\nu\) on \(\mathbb{C}(x,y)\) centered at infinity (i.e., \(\nu(f) < 0\) for some \(f \in \mathbb{C}[x,y]\)). The author introduces key forms for divisorial degrees as a counterpart of key polynomials of valuations. The main result in the paper provides a characterization of nonnegative (and positive) divisorial degrees in terms of key forms, playing and special role the last key form. The maps characterized in this paper have interesting properties involving the global geometry of those surfaces defined by the corresponding valuations and can be regarded as the natural extension of negatives of divisorial valuations whose general elements are given by curves with only one place at infinity. In the paper, it is also considered an extended value semigroup of the semidegree and is related to the fact that key forms be polynomials and the determination of algebraic compactifications of \(\mathbb{C}^2\).