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<p>A polynomial <i>f</i> ∈ ℂ[<i>x,y</i>] is a Jacobian mate if the Jacobian J(<i>f, g</i>) = 1 for some <i>g</i> ∈ ℂ[<i>x,y</i>]. It is not known that then ℂ[<i>f,g</i>]= ℂ[<i>x,y</i>], and a conjecture that this is the case is the Jacobian conjecture. In this note we will check that if <i>f</i> is given then, the subalgebra ℂ[<i>f, g</i>] is known and that <i>g</i> can be recovered up to asummand which is a polynomial in <i>f</i>.</p>

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A Jacobian mate defines the Jacobian pair

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Leonid Makar-Limanov

摘要

A polynomial f ∈ ℂ[x,y] is a Jacobian mate if the Jacobian J(f, g) = 1 for some g ∈ ℂ[x,y]. It is not known that then ℂ[f,g]= ℂ[x,y], and a conjecture that this is the case is the Jacobian conjecture. In this note we will check that if f is given then, the subalgebra ℂ[f, g] is known and that g can be recovered up to asummand which is a polynomial in f.