2010/06/30 by Arno van den Essen, Essen, Arno van den
Mathematics · #14E05 #14R15 #16S32 #33C45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:14E05 #msc:14R15 #msc:16S32 #msc:33C45
paper · pdf · doi:10.48550/arxiv.1006.5801
Latex, 24 pages
arxiv created 2010/06/30 · openalex publication_date 2010/06/30 · arxiv updated 2010/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we discuss a general framework in which we present a new conjecture, due to Wenhua Zhao, the Image Conjecture. This conjecture implies the Generalized Vanishing Conjecture and hence the Jacobian Conjecture. Crucial ingredient is the notion of a Mathieu space: let k be a field and R a commutative k-algebra. A k-linear subspace M of R is called a Mathieu subspace of R, if the following holds: let f∈ R be such that fm∈ M, for all m≥ 1, then for every g∈ R also gfm∈ M, for almost all m, i.e. only finitely many exceptions. Let A be the polynomial ring in ζ=ζ1, ...,ζn and z1, ...,zn over \mathbb C. The Image Conjecture (IC) asserts that ∑i(∂zi-ζi)A is a Mathieu subspace of A. We prove this conjecture for n=1. Also we relate (IC) to the following Integral Conjecture: if B is an open subset of \mathbb Rn and σ a positive measure, such that the integral over B of each polynomial in z over \mathbb C is finite, then the set of polynomials, whose integral over B is zero, is a Mathieu subspace of \mathbb C[z]. It turns out that Laguerre polynomials play a special role in the study of the Jacobian Conjecture.