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New reduction in the Jacobian conjecture.

Drużkowski, Ludwik M. — 2001

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

An Effective Approach to Keller's Jacobian Conjecture.

Ludwik M. Druzkowski — 1983

Mathematische Annalen

A generalization of the Malgrange-Zerner theorem

Ludwik M. Drużkowski — 1980

Annales Polonici Mathematici

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

Ludwik M. Drużkowski — 1997

Annales Polonici Mathematici

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F (x ₁, . . ., x_{n}) = x - H (x) : = (x ₁ - H ₁ (x ₁, . . ., x_{n}), . . ., x_{n} - H_{n} (x ₁, . . ., x_{n}))$ , 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 $x = (x ₁, . . ., x_{n}) \in ℝ^{n}$ . 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 $d e g F^{- 1} \leq {(d e g F)}^{i n d F - 1}$ , where $i n d F : = m a x i n d H^{'} (x) : x \in ℝ^{n}$ . Note that the above inequality is not true when the coefficients of...

A geometric approach to the Jacobian Conjecture in ℂ²

Ludwik M. Drużkowski — 1991

Annales Polonici Mathematici

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.

Extension of separately holomorphic functions defined in non-open sets in the infinite dimensional case

Ludwik M. Drużkowski — 1983

Annales Polonici Mathematici

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

Ludwik M. Drużkowski — 2005

Annales Polonici Mathematici

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...

Differential conditions to verify the Jacobian Conjecture

Ludwik M. Drużkowski; Halszka K. Tutaj — 1992

Annales Polonici Mathematici

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.

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Currently displaying 1 – 8 of 8

New reduction in the Jacobian conjecture.

An Effective Approach to Keller's Jacobian Conjecture.

A generalization of the Malgrange-Zerner theorem

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

A geometric approach to the Jacobian Conjecture in ℂ²

Extension of separately holomorphic functions defined in non-open sets in the infinite dimensional case

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

Differential conditions to verify the Jacobian Conjecture