On the shape of possible counterexamples to the Jacobian Conjecture

2014/01/08 by Jorge A. Guccione, Guccione, Jorge A., Juan J. Guccione +3 · 2 citations
Mathematics · #13F20 #14R15 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1401.1784

openalex publication_date 2014/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. We give an elementary proof of the result of Heitmann, which states that gcd(deg(P),deg(Q)) is greater than or equal to 16 for any counterexample (P,Q). We also prove that gcd(deg(P),deg(Q)) ≠ 2p for any prime p and analyze thoroughly the case 16, adapting a reduction of degree technique introduced by Moh.

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• Normal forms for ordinary differential operators, II
• The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture

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