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An ideal-theoretic approach to Keller maps
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 4 / November 2019
- Published online by Cambridge University Press:
- 11 June 2019, pp. 1033-1044
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- Article
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A self-map F of an affine space
${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in 
$k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.
