scholarly journals Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture

1996 ◽  
Vol 247 ◽  
pp. 121-132 ◽  
Author(s):  
Arno van den Essen ◽  
Engelbert Hubbers
Keyword(s):  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Polynomial Maps ◽  
Strongly Nilpotent

scholarly journals Irreducibility Properties of Keller Maps

2016 ◽  
Vol 23 (04) ◽  
pp. 663-680 ◽  
Author(s):  
Michiel de Bondt ◽  
Dan Yan
Keyword(s):  
Linear Combination ◽  
Characteristic Zero ◽  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Extra Condition ◽  
Irreducible Polynomials ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Keller Map

Jędrzejewicz 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 irreducibility 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.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO

2018 ◽  
Vol 98 (1) ◽  
pp. 89-101 ◽  
Author(s):  
MICHIEL DE BONDT ◽  
XIAOSONG SUN
Keyword(s):  
Homogeneous Polynomial ◽  
Jacobian Matrix ◽  
Jacobian Matrices ◽  
Polynomial Maps ◽  
Keller Map

Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.

An effective study of polynomial maps

2016 ◽  
Vol 16 (08) ◽  
pp. 1750141 ◽  
Author(s):  
Elżbieta Adamus ◽  
Paweł Bogdan ◽  
Teresa Crespo ◽  
Zbigniew Hajto
Keyword(s):  
Affine Space ◽  
Finite Sequence ◽  
Jacobian Conjecture ◽  
Equivalent Condition ◽  
Effective Algorithm ◽  
Polynomial Maps ◽  
Dimension Formula ◽  
Polynomial Map ◽  
Equivalent Statement

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

scholarly journals Symmetric Jacobians

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Michiel Bondt
Keyword(s):  
Jacobian Conjecture ◽  
Arbitrary Degree ◽  
Polynomial Maps

AbstractThis article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.

scholarly journals Injective polynomial maps and the Jacobian conjecture

1983 ◽  
Vol 28 (3) ◽  
pp. 235-239 ◽  
Author(s):  
E. Connell ◽  
L. van den Dries
Keyword(s):  
Jacobian Conjecture ◽  
Polynomial Maps

A Mountain Pass to the Jacobian Conjecture

1998 ◽  
Vol 41 (4) ◽  
pp. 442-451 ◽  
Author(s):  
Marc Chamberland ◽  
Gary Meisters
Keyword(s):  
Mountain Pass ◽  
Jacobian Conjecture ◽  
Mountain Pass Lemma ◽  
Sufficient Condition ◽  
Polynomial Maps ◽  
Class C ◽  
Open Question ◽  
Uniformly Bounded

AbstractThis paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem 1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of are uniformly bounded away from zero for x ∊ Rn, where is a class C1 map, then F is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class C1 maps F of Rn into itself is that all the eigenvalues of F′(x) are bounded away from zero on Rn. This is stated as Conjecture 2.1. If true, it would imply (via Reduction-of-Degree) injectivity of polynomial mapssatisfying the hypothesis, det F′(x) ≡ 1, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

scholarly journals An extension to the planar Markus–Yamabe Jacobian conjecture

2021 ◽  
pp. 1-6
Author(s):  
Marco Sabatini
Keyword(s):  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Differential Systems

We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts.

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