Michiel de Bondt

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).

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

The strong nilpotency index of a matrix

It is known that strongly nilpotent matrices over a division ring are linearly triangularizable. We describe the structure of such matrices in terms of the strong nilpotency index. We apply our results on quasi-translation such that has strong nilpotency index two.It is known that strongly nilpotent matrices over a division ring are linearly triangularizable. We describe the structure of such matrices in terms of the strong nilpotency index. We apply our results on quasi-translation such that has strong nilpotency index two.

Irreducibility properties of Keller maps

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

Mathieu subspaces of codimension less than n of Mat(n) (K)

We classify all Mathieu subspaces of of codimension less than n, under the assumption that or . More precisely, we show that any proper Mathieu subspace of of codimension less than n is a subspace of , if or . On the other hand, we show that every subspace of of codimension less than n in is a Mathieu subspace of , if or .We classify all Mathieu subspaces of of codimension less than n, under the assumption that or . More precisely, we show that any proper Mathieu subspace of of codimension less than n is a subspace of , if or . On the other hand, we show that every subspace of of codimension less than n in is a Mathieu subspace of , if or .

Some remarks on the Jacobian Conjecture and Drużkowski mappings

Abstract In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension r ⩾ 1 and give some partial results for r = 2 . Finally, for a homogeneous power linear Keller map F = x + H of degree d ⩾ 2 , we give the inverse polynomial map under the condition that J H 3 = 0 . We shall show that deg ( F − 1 ) ⩽ d k if k ⩽ 2 and J H k + 1 = 0 , but also give an example with d = 2 and J H 4 = 0 such that deg ( F − 1 ) > d 3 .

Polynomial maps with invertible sums of Jacobian matrices and directional derivatives

Abstract Let F : C n → C m be a polynomial map with deg F = d ≥ 2 . We prove that F is invertible if m = n and ∑ i = 1 d − 1 ( J F ) | α i is invertible for all α i ∈ C n , which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines L = { β + μ γ ∣ μ ∈ C } ⊆ C n ( γ ≠ 0 ), F ∣ L is linearly rectifiable, if and only if ∑ i = 1 d − 1 ( J F ) | α i ⋅ γ ≠ 0 for all α i ∈ L . This appears to be the case for all affine lines L when F is injective and d ≤ 3 . We also prove that if m = n and ∑ i = 1 n ( J F ) | α i is invertible for all α i ∈ C n , then F is a composition of an invertible linear map and an invertible polynomial map X + H with linear part X , such that the subspace generated by { ( J H ) | α ∣ α ∈ C n } consists of nilpotent matrices.

Homogeneous quasi-translations in dimension 5

We give a proof in modern language of the following result by Paul Gordan and Max Nöther: a homogeneous quasi-translation in dimension 5 without linear invariants would be linearly conjugate to another such quasi-translation

Some remarks on the Jacobian conjecture and polynomial endomorphisms

x + H