The Maximum Modulus Set of a Polynomial

Pardo-Simón, Leticia and Sixsmith, David J. (2022). The Maximum Modulus Set of a Polynomial. Computational Methods and Function Theory, 22 pp. 207–214.

DOI: https://doi.org/10.1007/s40315-021-00368-7

Abstract

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 p̃ 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,…, α_n of distinct positive real numbers, we construct polynomials p and p̃ such that M(p) has discontinuities of modulus α₁, α₂,…, α_n, and M(p̃) has singleton components at the points α₁, α₂,…, α_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.

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