On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

2022/01/17 by William E. Hurst, Hurst, William E., Kyungyong Lee +5
Computer Science · Mathematics · #11P21 #14M25 #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 #Primary: 14R15 Secondary: 13F20

paper · pdf · doi:10.48550/arxiv.2201.06613

openalex publication_date 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be an algebraically closed field of characteristic 0. When the Jacobian (∂ f/∂ x)(∂ g/∂ y) - (∂ g/∂ x)(∂ f/∂ y) is a constant for f,g∈ K[x,y], Magnus' formula from [A. Magnus, Volume preserving transformations in several complex variables, Proc. Amer. Math. Soc. 5 (1954), 256--266] describes the relations between the homogeneous degree pieces f_i's and g_i's. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.

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