We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.
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Let F = X + H be a cubic homogeneous polynomial automorphism from $ℂ^n$ to $ℂ^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg F^{-1} ≤ 3^{p-1}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.
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The tame generators problem asked if every invertible polynomial map is tame, i.e. a finite composition of so-called elementary maps. Recently in [8] it was shown that the classical Nagata automorphism in dimension 3 is not tame. The proof is long and very technical. The aim of this paper is to present the main ideas of that proof.
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We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski's result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + (Ax)^{*3}$ with A² = 0. Then we describe the authors' result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f ∈ k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao's reformulation of the JC which asserts the following: for every homogeneous polynomial $f ∈ k^{[n]}$ (of degree 4) the hypothesis $Δ^m(f^m) = 0$ for all m ≥ 1 implies that $Δ^{m-1}(f^m) = 0$ for all large m (Δ is the Laplace operator). In the last section we describe Kumar's formulation of the JC in terms of smoothness of a certain family of hypersurfaces.
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Let k be an algebraically closed field of characteristic zero and $F:= x + (Ax)^{*d}: kⁿ → kⁿ$ a Drużkowski mapping of degree ≥ 2 with det JF = 1. We classify all such mappings whose Jacobian matrix JF is symmetric. It follows that the Jacobian Conjecture holds for these mappings.
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