On lower bounds for the complexity of polynomials and their multiples

Abstract.

We prove a theorem giving arbitrarily long explicit sequences $x_{1},\ldots,x_{s}$ of algebraic numbers such that any nonzero polynomial f(X) satisfying $f(x_{1}) = \cdots = f(x_{s}) = 0$ has nonscalar complexity $> C \sqrt{s}$ for some positive constant C independent of s. A similar result is shown for rapidly growing rational sequences.