On extending Abhyankar's two point lemma to positive weights

Abstract

In this paper we answer the question of whether we can extend the current proof of Abhyankar's two point lemma to any positive weight in the negative. This is done by constructing suitable $w$-homogeneous polynomial pairs of positive weight that behave the same way under the Jacobian as their negatively weighted counterparts, but they do not possess the same pairwise properties.