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A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski (1998)

Colloquium Mathematicae

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 ) = ∑ 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 large d....

Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, W ⊂ k n + 1 , and F : k ⁿ → k n + 1 such that F | V = f and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, W ⊂ k m with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

Global minimal models for endomorphisms of projective space

Clayton Petsche, Brian Stout (2014)

Journal de Théorie des Nombres de Bordeaux

We prove the existence of global minimal models for endomorphisms φ : ℙ N → ℙ N of projective space defined over the field of fractions of a principal ideal domain.

Halphen pencils on weighted Fano threefold hypersurfaces

Ivan Cheltsov, Jihun Park (2009)

Open Mathematics

On a general quasismooth well-formed weighted hypersurface of degree Σi=14 a i in ℙ(1, a 1, a 2, a 3, a 4), we classify all pencils whose general members are surfaces of Kodaira dimension zero.

Jung's type theorem for polynomial transformations of ℂ²

Sławomir Kołodziej (1991)

Annales Polonici Mathematici

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.

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