An intersection theoretical proof of the embedding line theorem (Q1320236)

Embedding line theorem: Let \(K\) be an algebraically closed field of characteristic zero. If \(u(x,y) = 0\) defines an embedded affine line \(K\) in \(K^ 2\), then there exists a polynomial \(v(x,y) \in K [x,y]\) such that \(K[u,v] = K[x,y]\). A new proof of the embedding line theorem in characteristic zero is given. The computation is based on Bezout's theorem and Zariski's analysis [\textit{O. Zariski}, ``Le problème des modules pour les branches planes'' (Paris 1973; Zbl 0317.14004)] of the semigroup of intersection numbers at infinite place of the embedded line. The role of the approximate root is clarified from the intersection theoretical point of view.