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Effective bounds for the zeros of linear recurrences in function fields

Clemens Fuchs, Attila Pethő (2005)

Journal de Théorie des Nombres de Bordeaux

In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation G n ( x ) = G m ( P ( x ) ) , ( m , n ) 2 , where ( G n ( x ) ) is a linear recurring sequence of polynomials and P ( x ) is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

Factorisation d'opérateurs différentiels à coefficients dans une extension liouvillienne d'un corps valué

Magali Bouffet (2002)

Annales de l’institut Fourier

On démontre ici un lemme de Hensel pour les opérateurs différentiels. On en déduit un théorème de factorisation pour des opérateurs différentiels à coefficients dans une extension liouvillienne transcendante d’un corps valué. On obtient en particulier un théorème de factorisation pour des opérateurs différentiels à coefficients dans une extension de ( ( z ) ) par un nombre fini d’exponentielles et de logarithmes algébriquement indépendants sur ( ( z ) ) .

Finite automata and algebraic extensions of function fields

Kiran S. Kedlaya (2006)

Journal de Théorie des Nombres de Bordeaux

We give an automata-theoretic description of the algebraic closure of the rational function field 𝔽 q ( t ) over a finite field 𝔽 q , generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over 𝔽 q . In passing, we obtain a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...

Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

Yong HU (2012)

Annales de l’institut Fourier

Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let Ω R be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of Spec R . We prove that a quadratic form q over L satisfies the local-global principle with respect to Ω R in the following two cases: (1) q has rank 3 or 4; (2) q has rank 5 and R = A [ [ y ] ] , where A is a complete discrete valuation ring with...

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