A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen

Displaying similar documents to “A counterexample to a conjecture of Drużkowski and Rusek”

The Jacobian Conjecture: survey of some results

Ludwik Drużkowski (1995)

Banach Center Publications

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The paper contains the formulation of the problem and an almost up-to-date survey of some results in the area.

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.

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.

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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New reduction in the Jacobian conjecture.

Drużkowski, Ludwik M. (2001)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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Homogeneous polynomial invariants for cubic-homogeneous functions.

Zampieri, Gaetano (2008)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski (1998)

Colloquium Mathematicae

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Let F=X-H: $k^{n}$ → $k^{n}$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of $G_{i}$ of degree 2d+1 can be expressed as $G_{i}^{(d)} = \sum_{T} α {(T)}^{- 1} σ_{i} (T)$ , where T varies over rooted trees with d vertices, α(T)=CardAut(T) and $σ_{i} (T)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, $G_{i}^{(d)}$ is zero for sufficiently...

Thomas’ conjecture over function fields

Volker Ziegler (2007)

Journal de Théorie des Nombres de Bordeaux

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Thomas’ conjecture is, given monic polynomials $p_{1},$ $..., p_{d} \in ℤ [a]$ with $0 < deg p_{1} < \dots < deg p_{d}$ , then the Thue equation (over the rational integers) $(X - p_{1} (a) Y) \dots (X - p_{d} (a) Y) + Y^{d} = 1$ has only trivial solutions, provided $a \geq a_{0}$ with effective computable $a_{0}$ . We consider a function field analogue of Thomas’ conjecture in case of degree $d = 3$ . Moreover we find a counterexample to Thomas’ conjecture for $d = 3$ .

On the Euler characteristic of fibres of real polynomial maps

Adam Parusiński, Zbigniew Szafraniec (1998)

Banach Center Publications

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Let Y be a real algebraic subset of $^{m}$ and $F : Y \to^{n}$ be a polynomial map. We show that there exist real polynomial functions $g_{1}, . . ., g_{s}$ on $^{n}$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_{i}$ .