Arno van den Essen

An algorithm to compute the invariant ring of a G a -action on an affine variety

Abstract We describe an algorithm which computes the invariants of all G a -actions on affine varieties, in case the invariant ring is finitely generated. The algorithm is based on a study of the kernel of a locally nilpotent derivation and some algorithms from the theory of Grobner bases.

A reduction of the Jacobian Conjecture to the symmetric case

The main result of this paper asserts that it suffices to prove the Jacobian Conjecture for all polynomial maps of the form x + H, where H is homogeneous (of degree 3) and JH is nilpotent and symmetric. Also a 6-dimensional counterexample is given to a dependence problem posed by de Bondt and van den Essen (2003).

Some combinatorial questions about polynomial mappings

Abstract The purpose of this note is to show how recent progress in non-commutative combinatorial algebra gives a new input to Jacobian-related problems in a commutative situation.

Locally nilpotent derivations and their applications, III

We give an algorithm to decide if certain derivations are locally nilpotent. This algorithm is used to give a solution of the extendability problem for n≤3 (i.e. canF1, …, Fn−1 be extended to a polynomial automorphism (F1, …, Fn−1, Fn):Cn → Cn) and to indicate a strategy for solving the n-dimensional extendability problem. We also reformulate the cancellation problem in terms of locally nilpotent derivations.

The Jacobian Conjecture: Linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F : k 3 → k 3 a polynomial map of the form F = x + H , where H is homogeneous of degree d 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T −1 HT = (0 ,h 2(x1), h3(x1 ,x 2)) ,w here thehi are homogeneous of degree d .A s a consequence of this result, we show that all generalized Druu zkowski mappings F = x + H = (x1 + L d ,...,x n + L d) ,w hereLi are linear forms for all i and d 2, are linearly triangularizable

Seven Lectures on Polynomial Automorphisms

Throughout these lectures we use the following notation and terminology: ℕ:= {1, 2, 3,...}, ℕ = ℕ ∪ {0}, ℚ = the rational numbers, ℝ:= the real numbers and ℂ:= the complex numbers. Furthermore k will denote an arbitrary field and F = (F 1, ..., F n): k n → k n a polynomial map i.e. a map of the form

Triangular derivations related to problems on affine n-space

\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right) \mapsto \left( {{{F}_{1}}\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right), \ldots ,{{F}_{n}}\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)} \right),

Derivations having divergence zero on R[X,Y]

where each F i belongs to the polynomial ring k[X]: = k[X 1, ..., X n ].