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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....

A primrose path from Krull to Zorn

Marcel Erné (1995)

Commentationes Mathematicae Universitatis Carolinae

Given a set X of “indeterminates” and a field F , an ideal in the polynomial ring R = F [ X ] is called conservative if it contains with any polynomial all of its monomials. The map S ↦ R S yields an isomorphism between the power set P ( X ) and the complete lattice of all conservative prime ideals of R . Moreover, the members of any system S ⊆ P ( X ) of finite character are in one-to-one correspondence with the conservative prime ideals contained in P S = ⋃ { R S : S ∈ S } , and the maximal members of S correspond to the maximal ideals contained in...

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