On the Number of Additions to Compute Specific Polynomials

The number of addition-subtraction operations required to compute univariate pol nomials is investigated. The existence of rational coefficient polynomials of degree n requiring $ \sim (\sqrt n ) \pm $ operations is established using an argument based on algebraic independence. A more analytic argument is used to relate $ \pm $ complexity to the number of distinct real zeros possessed by a given real coefficient polynomial.