On extending Abhyankar's two point lemma to positive weights
Abhyankar proved that if \(f,g\in \mathbb{K}[X,Y]\) and \(\text{Jac}(f,g)= \text{const}\neq 0\) then for any negative weight \(w=(w_{1},w_{2})\) (i.e. \( (w_{1},w_{2})\in (\mathbb{Z}\times \mathbb{Z)}^{\ast },\) \(\gcd (w_{1},w_{2})=1\) and \(\left( -w_{2}/w_{1}\right) <0\)) then \(f\) has at most two \(w\)-points at infinity. The author shows that it does not hold for positive weights by giving counterexamples (in case positive weights) to two lemmas Abhyankar used in the proof of his theorem.
scientific article; zbMATH DE number 6339721
scientific article