  The degree sequence of the algebraic numbers in an algebraic linear
recurrence sequence is shown to be virtually periodic. This is proved using the
Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the
minimal polynomial sequence for exponential sums over finite fields. The degree
periodicity also holds for some more complicated non-linear recurrence
sequences. We give one example from the iterations of a polynomial map. This
depending on the dynamic Mordell-Lang conjecture which has been proved in some
cases.
