On extending Abhyankar's two point lemma to positive weights
type: Item
instance of: scholarly article
title: On extending Abhyankar's two point lemma to positive weights
zbMATH Open document ID: 1308.14065
DOI: 10.1216/RMJ-2014-44-2-685
author: James C. Price
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1407154919
review text: 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.
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 14R15
zbMATH DE Number: 6339721
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: Abhyankar's two point lemma
MaRDI profile type: Publication
Wikidata QID: Q124811306
cites work: Lectures on expansion techniques in algebraic geometry. With notes by Balwant Singh
cites work: Some remarks on the Jacobian question. (Notes by Marius van der Put and William Heinzer, updated by Avinash Sathaye)
cites work: Some thoughts on the Jacobian conjecture. I
cites work: Some thoughts on the Jacobian conjecture. II
cites work: Some thoughts on the Jacobian conjecture. III
cites work: The Jacobian conjecture in two variables
cites work: On polynomial Solutions of a differential equation
cites work: Q3497182
cites work: Q3033874
cites work: A generalization of Magnus' theorem
cites work: On the boundary obstructions to the Jacobian problem
cites work: Q4369146
description: scientific article; zbMATH DE number 6339721