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Galois theory of q -difference equations

Marius van der Put, Marc Reversat (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Choose q ∈ ℂ with 0 < | q | < 1 . The main theme of this paper is the study of linear q -difference equations over the field K of germs of meromorphic functions at 0 . A systematic treatment of classification and moduli is developed. It turns out that a difference module M over K induces in a functorial way a vector bundle v ( M ) on the Tate curve E q : = ℂ * / q ℤ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification ( [ A t ] ) of the indecomposable vector bundles on the complex Tate...

Galois theory of special trinomials.

Shreeram S. Abhyankar (2003)

Revista Matemática Iberoamericana

This is the material which I presented at the 60th birthday conference of my good friend José Luis Vicente in Seville in September 2001. It is based on the nine lectures, now called sections, which were given by me at Purdue in Spring 1997. This should provide a good calculational background for the Galois theory of vectorial ( = additive) polynomials and their iterates.

Generalization of Ehrlich-Kjurkchiev Method for Multiple Roots of Algebraic Equations

Iliev, Anton (1998)

Serdica Mathematical Journal

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.

Generalized Kummer theory and its applications

Toru Komatsu (2009)

Annales mathématiques Blaise Pascal

In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ ∉ k and ω ∈ k where ζ is a primitive n -th root of unity and ω = ζ + ζ - 1 . In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.

Generalized Staircases: Recurrence and Symmetry

W. Patrick Hooper, Barak Weiss (2012)

Annales de l’institut Fourier

We study infinite translation surfaces which are ℤ -covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

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