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The Strong Anick Conjecture is true

Vesselin Drensky, Jie-Tai Yu (2007)

Journal of the European Mathematical Society

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K 〈 x , y , z 〉 over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K 〈 x , y , z 〉 . In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a...

The valuated ring of the arithmetical functions as a power series ring

Emil Daniel Schwab, Gheorghe Silberberg (2001)

Archivum Mathematicum

The paper examines the ring A of arithmetical functions, identifying it to the domain of formal power series over 𝐂 in a countable set of indeterminates. It is proven that A is a complete ultrametric space and all its continuous endomorphisms are described. It is also proven that A is a quasi-noetherian ring.

Théorèmes de préparation Gevrey et étude de certaines applications formelles

Augustin Mouze (2003)

Annales Polonici Mathematici

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps F from ℂ s to ℂ p without constant term such that the rank of the Jacobian matrix of F is equal to 1. Let be a formal power series. If F is a holomorphic map, the following result is well known: ∘ F is analytic implies there exists a convergent power series...

Topological characterizations of ordered groups with quasi-divisor theory

Jiří Močkoř (2002)

Czechoslovak Mathematical Journal

For an order embedding G → h → Γ of a partly ordered group G into an l -group Γ a topology 𝒯 W ^ is introduced on Γ which is defined by a family of valuations W on G . Some density properties of sets h ( G ) , h ( X t ) and ( h ( X t ) ∖ { h ( g 1 ) , ⋯ , h ( g n ) } ) ( X t being t -ideals in G ) in the topological space ( Γ , 𝒯 W ^ ) are then investigated, each of them being equivalent to the statement that h is a strong theory of quasi-divisors.

Totally indefinite Euclidean quaternion fields

Jean-Paul Cerri, Jérôme Chaubert, Pierre Lezowski (2014)

Acta Arithmetica

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.

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