On the roots of certain polynomials arising from the analysis of the Nelder–Mead simplex method

Abstract The study of the effect of dimensionality on the Nelder–Mead simplex method for unconstrained optimization leads us to the study of a two parameter family of polynomials of the form p n ( z )= b − az −⋯− az n −1 + z n . We show that provided that a −a b is real, it is possible to use, primarily, the Schur–Cohn Criterion in order to determine the configuration of the roots of p n ( z ) with respect to the unit circle.