On the multiplicity of linear recurrence sequences
نویسنده
چکیده
We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences. 1 Linear recurrence sequences Let {un}n∈Z be a linear recurrence sequence of complex numbers satisfying the recurrence relation un = c1un−1 + · · ·+ ctun−t, (1) for c1, . . . , ct ∈ C with ct 6= 0. We say it is of order t if it satisfies (1) but no such relation with fewer than t summands. We define the companion polynomial of our recurrence sequence by P(z) = z − c1z − · · · − ct. Say our companion polynomial factors over C as
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