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.

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