The Maximum Modulus Set of a Polynomial

Pardo-Simón, Leticia and Sixsmith, David J. (2021). The Maximum Modulus Set of a Polynomial. Computational Methods and Function Theory (Early Access).



We study the maximum modulus set, M(p), of a polynomial p. We are interested in constructing p so that M(p) has certain exceptional features. Jassim and London gave a cubic polynomial p such that M(p) has one discontinuity, and Tyler found a quintic polynomial such that M(p̃) has one singleton component. These are the only results of this type, and we strengthen them considerably. In particular, given a finite sequence α1, α2,…, αn of distinct positive real numbers, we construct polynomials p and such that M(p) has discontinuities of modulus α1, α2,…, αn, and M(p̃) has singleton components at the points α1, α2,…, αn. 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.

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