Abstract
AbstractLet $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.
Highlights
Ltioetn(GGnn)=∞n=a0 1b(en)aαnn1o+nd·e·g· e+neart(ante)αlintn.eTahrirseecxuprrreensscieosnemquaeknecseswenitshe power sum representafor a sequence (Gn)∞n=0 taking values in any field K; the characteristic roots αi as well as the coefficients of the polynomials ai lie in a finite extension L of K
We prove a function field analogue of the well-known result in is given by the number field case that, under some nonrestrictive conditions, |Gn| ≥n(1−ε) for n large enough
The nondegenerate condition means in the number field case that no ratio αi/αj for i j is a root of unity and in the function field case that no ratio αi/αj for i j is contained in the field of constants
Summary
Ltioetn(GGnn)=∞n=a0 1b(en)aαnn1o+nd·e·g· e+neart(ante)αlintn.eTahrirseecxuprrreensscieosnemquaeknecseswenitshe power sum representafor a sequence (Gn)∞n=0 taking values in any field K; the characteristic roots αi as well as the coefficients of the polynomials ai lie in a finite extension L of K. We will prove a theorem which states an inequality for an arbitrary valuation in the splitting field L of the characteristic polynomial belonging to the linear recurrence sequence. Mj k=0 ajk nk and any valuation μ (in a function field L/K containing the αj and the coefficients of the aj) we Downloaded from https://www.cambridge.org/core.
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