Polynomial maps with nilpotent Jacobian

Abstract

In the previous chapter we showed that every polynomial map whose Jacobian matrix is invertible is stably equivalent to a polynomial map of the form X + H, where J H is nilpotent. For polynomial maps over a field of characteristic zero, such maps with the additional assumption that H is cubic homogeneous were first studied by Wright in dimension 3, [294] and Hubbers in dimension 4, [156]. In both cases it turned out that the rows of the Jacobian matrix are linearly dependent over the coefficient field. This gave rise to the formulation of a (still open) dependence problem, 7.1.5. This problem led Hubbers and the author to the discovery of a large class of polynomial maps (over an arbitrary commutative ring A) having nilpotent Jacobian matrix. This class will be defined and studied in §2 and is denoted by H(n, A). So H(n, A) is a subset of N(n, A), the set of all polynomial maps H ∈ A [X]n with J H nilpotent. If A is a UFD of characteristic zero we will show that H(2, A) = N(2, A). However if n ≥ 3 the class N(n, A) is essentially larger than H(n, A). Furthermore we will show that all maps of the form X + H with H ∈ H(n, A) are polynomial automorphisms. In fact we show that every such map is a finite product of exponentials of a special kind of locally nilpotent derivations, the so-called nice derivations, 7.3.18, and we deduce that all these maps are stably tame. At the end of §3 we give a shorter proof of this stable tameness result due to Hubbers and Wright in [158].KeywordsInduction HypothesisJacobian MatrixCharacteristic ZeroMain DiagonalRing HomomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.