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1

A geometric approach to the Jacobian Conjecture in ℂ²

100%

Drużkowski L.

Annales Polonici Mathematici

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1991

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tom 55

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nr 1

95-101

EN

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set $g^{-1}(0)$ (resp. $f^{-1}(0)$), then (f,g) is bijective.

2

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

100%

Drużkowski L.

Annales Polonici Mathematici

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2005

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tom 87

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nr 1

83-92

EN

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 homogeneous form and prove (JC) for maps having cubic linear form with symmetric F'(x), more precisely: polynomial maps of cubic linear form with symmetric F'(x) and constant nonzero jacobian are tame automorphisms.

3

The Jacobian Conjecture in case of "non-negative coefficients"

100%

Drużkowski L.

Annales Polonici Mathematici

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1997

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tom 66

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nr 1

67-75

EN

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F(x₁,...,x_n) = x - H(x) := (x₁ - H₁(x₁,...,x_n),...,x_n - H_n(x₁,...,x_n))$, where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j = 0$), j = 1,...,n and H'(x) is a nilpotent matrix for each $x = (x₁,...,x_n) ∈ ℝ^n$. We give another proof of Yu's theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $deg F^{-1} ≤ (deg F)^{ind F - 1}$, where $ind F := max{ind H'(x): x ∈ ℝ^n}$. Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].

4

Differential conditions to verify the Jacobian Conjecture

63%

Drużkowski L., Tutaj H.

Annales Polonici Mathematici

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1992

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tom 57

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nr 3

253-263

EN

Let F be a polynomial mapping of ℝ², F(O) = 0. In 1987 Meisters and Olech proved that the solution y(·) = 0 of the autonomous system of differential equations ẏ = F(y) is globally asymptotically stable provided that the jacobian of F is everywhere positive and the trace of the matrix of the differential of F is everywhere negative. In particular, the mapping F is then injective. We give an n-dimensional generalization of this result.

Ograniczanie wyników

4 Annales Polonici Mathematici

4 Drużkowski L.

1 Tutaj H.

1 2005

1 1997

1 1992

1 1991

A geometric approach to the Jacobian Conjecture in ℂ²

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

The Jacobian Conjecture in case of "non-negative coefficients"

Differential conditions to verify the Jacobian Conjecture