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<p>A polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f \in \mathbb {C}[x,y]\)</EquationSource> </InlineEquation> is a Jacobian mate if the Jacobian <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\text{J}(f,g) = 1\)</EquationSource> </InlineEquation> for some <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(g \in \mathbb {C}[x,y]\)</EquationSource> </InlineEquation>. It is not known that then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {C}[f,g] = \mathbb {C}[x,y]\)</EquationSource> </InlineEquation> and a conjecture that this is the case is the Jacobian conjecture (JC). In this note we will assume that a counterexample to JC exists and obtain additional restrictions on <i>f</i>.</p>

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Properties of a Jacobian mate

  • 期刊论文

  • 发表日期：

    2026年4月30日

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  • Leonid Makar-Limanov,
  • Leonid Trakhtenberg

摘要

A polynomial \(f \in \mathbb {C}[x,y]\) is a Jacobian mate if the Jacobian \(\text{J}(f,g) = 1\) for some \(g \in \mathbb {C}[x,y]\) . It is not known that then \(\mathbb {C}[f,g] = \mathbb {C}[x,y]\) and a conjecture that this is the case is the Jacobian conjecture (JC). In this note we will assume that a counterexample to JC exists and obtain additional restrictions on f.