The distance between terms of an algebraic recurrence sequence.

§ 1. Let {wm}^=o be a sequence of algebraic numbers satisfying a recurrence relation with complex coefficients of order k, (!) wm4.k = v f c _ 1 w m + f c _ 1 + v k _ 2 w m + k _ 2 4 f v0wm, m = 0, 1,2, . . . , such that | w 0 | H \ u ± \ + ··· + |w k _ 1 | >0 and ν0Φθ. We assume that {Mm}^=0 does not satisfy a recurrence relation of lower order. Since the recurrence relation of minimal order is unique, this implies that the numbers v0, v l 9 . . . , v f c_ t are algebraic (cf. [5] §18). Let F be the field 0(w0, ul9... , w f c _ i , v0, v 1 ? . . . , ν^.^. Hence umeF for all m. Write the companion polynomial of (1) s