Displaying similar documents to “Reduction theorems for the Strong Real Jacobian Conjecture”

The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case

Ludwik M. Drużkowski (2005)

Annales Polonici Mathematici

Similarity:

Let 𝕂 denote ℝ or ℂ, n > 1. The Jacobian Conjecture can be formulated as follows: If F:𝕂ⁿ → 𝕂ⁿ is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism. Although the Jacobian Conjecture is still unsolved even in the case n = 2, it is convenient to consider the so-called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n>1. We present the reduction of (GJC) to the case of F of degree 3 and of symmetric...

Triangularization properties of power linear maps and the Structural Conjecture

Michiel de Bondt, Dan Yan (2014)

Annales Polonici Mathematici

Similarity:

We discuss several additional properties a power linear Keller map may have. The Structural Conjecture of Drużkowski (1983) asserts that certain two such properties are equivalent, but we show that one of them is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.

An update on a few permanent conjectures

Fuzhen Zhang (2016)

Special Matrices

Similarity:

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We...

Cores, Joins and the Fano-Flow Conjectures

Ligang Jin, Eckhard Steffen, Giuseppe Mazzuoccolo (2018)

Discussiones Mathematicae Graph Theory

Similarity:

The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud...

Plane Jacobian conjecture for simple polynomials

Nguyen Van Chau (2008)

Annales Polonici Mathematici

Similarity:

A non-zero constant Jacobian polynomial map F=(P,Q):ℂ² → ℂ² has a polynomial inverse if the component P is a simple polynomial, i.e. its regular extension to a morphism p:X → ℙ¹ in a compactification X of ℂ² has the following property: the restriction of p to each irreducible component C of the compactification divisor D = X-ℂ² is of degree 0 or 1.

A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen (1995)

Annales Polonici Mathematici

Similarity:

Let F = X + H be a cubic homogeneous polynomial automorphism from ℂ n to ℂ n . Let p be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that d e g F - 1 ≤ 3 p - 1 . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.