The Maximum Modulus Set of a Polynomial

Abstract

AbstractWe study the maximum modulus set, $${{\mathcal {M}}}(p)$$ M ( p ) , of a polynomial p. We are interested in constructing p so that $${{\mathcal {M}}}(p)$$ M ( p ) has certain exceptional features. Jassim and London gave a cubic polynomial p such that $${{\mathcal {M}}}(p)$$ M ( p ) has one discontinuity, and Tyler found a quintic polynomial $${\tilde{p}}$$ p ~ such that $${{\mathcal {M}}}({\tilde{p}})$$ M ( p ~ ) has one singleton component. These are the only results of this type, and we strengthen them considerably. In particular, given a finite sequence $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n of distinct positive real numbers, we construct polynomials p and $${\tilde{p}}$$ p ~ such that $${{\mathcal {M}}}(p)$$ M ( p ) has discontinuities of modulus $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n , and $${{\mathcal {M}}}({\tilde{p}})$$ M ( p ~ ) has singleton components at the points $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n . Finally we show that these results are strong, in the sense that it is not possible for a polynomial to have infinitely many discontinuities in its maximum modulus set.