Propriétés algébriques de suites différentiellement finies

Linear recurring sequences over rings and modules

“…Definition. If, for a sequence u 9 p(k), there exists a finite algebraic extension Q of the field P such that for some m 9 hi0 k the sequence u has the representation (10.8), where c~ 9 Q and a,0,.., a~m,-1 9 Q* are distinct elements for any r E 1, k, then we say that u is an exponentially represented sequence [78]. The set of all such sequences is denoted by EP (k).…”

Algebraic and algorithmic aspects of linear difference equations

“…We give an application of Proposition 5.6 and an application of Corollary 5.8 in this section. The following result was conjectured in [3] and proven, when k = C, an algebraically closed field in [2] and [20]. The result and proof appear as Proposition 3.5 of [27].…”

“…Benzaghou/Bézivin[3]) Let C be an algebraically closed field and k a difference field with C ⊂ k ⊂ S C . Let u ∈ S C satisfy a linear difference equation over k and also satisfy a nonzero polynomial equation over k. Then u is the interlacing of sequences, each of which lies in a finite algebraic difference field extension of k. If k = C(x), σ(x) = x + 1, then these elements lie in C(x).…”

“…Let R be a PV extension of k with PV group Gal σ (R/k) G(C) ⊂ GL n (C). Then R is the coordinate ring of a k-torsor V for G. Furthermore the Galois action of φ ∈ Gal σ (R/k) on R is given by action ρ *[φ] on R induced by the action of [φ] ∈ G(C) on V via right multiplication as in(3).…”