2002/09/13 by S. S. Abhyankar, Shreeram S. Abhyankar, Abhyankar, S. S. +3
Computer Science · Mathematics · #12F10 #14H30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:12F10 #msc:14H30
paper · pdf · doi:10.48550/arxiv.math/0209159
latex, 41 pages
arxiv created 2002/09/13 · openalex publication_date 2002/09/13 · arxiv updated 2009/11/30 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We study meromorphic jacobian pairs, i.e., pairs of polynomials in one variable, with coefficients meromorphic series in a second variable, whose jacobian relative to the two variables depends only on the second variable. We pose two meromorphic jacobian conjectures about such pairs, one of which is in terms of an invariant of the pair which we call the beta invariant. These conjectures are shown to imply the bivariate algebraic jacobian conjecture which predicts that two bivariate polynomials generate the polynomial ring if their jacobian is a nonzero constant. As another technique for studying the jacobian conjecture we revisit the Newton polygon.