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PreprintCC BY 4.0arXiv

Boundary and Interior of Root Sets of Power Series

  • Jacob Kewarth
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Abstract

In this paper we study the set ℳM of roots of power series with coefficients in {0, ± 1, ..., ± M } and constant coefficient $1$. We prove that every nonreal root in the unit disk of a polynomial of this form is an interior point of ℳM. Moreover we establish a dimension formula and show the corresponding Hausdorff measure is finite and positive for a family of attractors parameterized by boundary parameters λ∈ ∂ ℳM ∖ ℝ. Finally we show that ∂ ℳM has Lebesgue measure zero.

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Cite

BibTeXRIS
BibTeX
@misc{kewarth2026boundary,
  title = {{Boundary and Interior of Root Sets of Power Series}},
  author = {Jacob Kewarth},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36279},
  eprint = {2610.36279},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36279},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Boundary and Interior of Root Sets of Power Series
AU  - Jacob Kewarth
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36279
UR  - https://arxiv.org/abs/2610.36279
AB  - In this paper we study the set ℳ_M of roots of power series with coefficients in {0, ± 1, ..., ± M } and constant coefficient $1$. We prove that every nonreal root in the unit disk of a polynomial of this form is an interior point of ℳ_M. Moreover we establish a dimension formula and show the corresponding Hausdorff measure is finite and positive for a family of attractors parameterized by boundary parameters λ∈ ∂ ℳ_M ∖ ℝ. Finally we show that ∂ ℳ_M has Lebesgue measure zero.
ER  -