The Jacobian Conjecture: survey of some results

Ludwik Drużkowski

Displaying similar documents to “The Jacobian Conjecture: survey of some results”

New reduction in the Jacobian conjecture.

Drużkowski, Ludwik M. (2001)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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A geometric approach to the Jacobian Conjecture in ℂ²

Ludwik M. Drużkowski (1991)

Annales Polonici Mathematici

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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.

A note on the limit points associated withthe generalized abc-conjecture for ℤ[t]

Daniel Davies (1996)

Colloquium Mathematicae

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Jung's type theorem for polynomial transformations of ℂ²

Sławomir Kołodziej (1991)

Annales Polonici Mathematici

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We prove that among counterexamples to the Jacobian Conjecture, if there are any, we can find one of lowest degree, the coordinates of which have the form $x^{m} y^{n}$ + terms of degree < m+n.

The degree of the inverse of a polynomial automorphism.

Brusadin, Sabrina, Gorni, Gianluca (2006)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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A simple counterexample to Kouchnirenko's conjecture.

Haas, Bertrand (2002)

Beiträge zur Algebra und Geometrie

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A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen (1995)

Annales Polonici Mathematici

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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 $d e g F^{- 1} \leq 3^{p - 1}$ . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

On a limit point associated with the abc-conjecture

Michael Filaseta, Sergeĭ Konyagin (1998)

Colloquium Mathematicae

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A finite dimensional reduction of the Schauder Conjecture

Espedito De Pascale (1993)

Commentationes Mathematicae Universitatis Carolinae

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Schauder’s Conjecture (i.eėvery compact convex set in a Hausdorff topological vector space has the f.p.p.) is reduced to the search for fixed points of suitable multivalued maps in finite dimensional spaces.

A consequence of an effective form of the abc-conjecture

Jerzy Browkin (1999)

Colloquium Mathematicae

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T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A $A^{0.4}$ with two exceptions given in Table 2.