scholarly journals A remark about homogeneous polynomial maps

2002 ◽  
Vol 19 (2) ◽  
pp. 257 ◽  
Author(s):  
Alexey A. Tret'yakov ◽  
Henryk Żołądek
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps

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 $n\neq 4$.

scholarly journals Commuting pairs of endomorphisms of

2016 ◽  
Vol 38 (3) ◽  
pp. 1025-1047
Author(s):  
LUCAS KAUFMANN
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps ◽  
Disjoint Sequence ◽  
Commuting Pairs ◽  
Number Of Iterations

We consider commuting pairs of holomorphic endomorphisms of $\mathbb{P}^{2}$ with disjoint sequence of iterates. The case that has not been completely studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of $\mathbb{C}^{2}$ inducing a Lattès map on the line at infinity.

On the Strong Nilpotence Problem

2014 ◽  
Vol 21 (01) ◽  
pp. 117-128 ◽  
Author(s):  
Xiaosong Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps ◽  
Strongly Nilpotent

In this paper, we describe the structure of quadratic homogeneous polynomial maps F=X+H with JH3=0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F=X+H is linearly triangularizable.

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 THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

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