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June 12, 2026Forum of Mathematics SigmaOpen Access

Shapiro’s problem on polynomials with large partial sums of coefficients

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Authors

MTMarc Technau

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Overview

Examines coefficient sums in polynomials, addressing Shapiro's problem on bounds and inequalities.

Key Points

  • To explore the maximum size of the sum of coefficients of polynomials bounded by one in the unit disk.
  • Investigated exact answers for related coefficient sums and asymptotic bounds for Shapiro's problem.
  • Applied a quantitative Eneström–Kakeya theorem for asymptotic analysis as n relates to d.
  • Developed inequalities for more generalized cases using Lagrange interpolators.
  • Provided an asymptotic answer to Shapiro’s problem when n is sufficiently small relative to d.
  • Established inequalities for coefficient sums that can give non-trivial bounds under specific conditions.
  • Identified cases where the inequality approaches sharpness.

Cite This Study

Marc Technau (2026) studied this question.

synapsesocial.com/papers/6a2ba5068101cf8926f032e0https://doi.org/10.1017/fms.2026.10213
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