Regular closedness off the real axis for root closures of bounded-height integer polynomials
Abstract
For each integer h≥1, let Zh be the closure in the complex plane of the roots of nonzero integer polynomials whose coefficients have absolute value at most h, with no degree restriction. We prove that Zh is regular closed off the real axis: every non-real point of Zh is a limit of interior points. The root-closure identification of Espigulé and Juher, for which we include a self-contained proof, shows that the same closure arises from monic polynomials and that outside the closed unit disc it is the connectedness locus of h+1 equally spaced collinear similarities. The main step proves that this locus is regular closed off the real axis; this was known for h=1 by Calegari, Koch and Walker, for h≥19, and inside an explicit lens. The symmetry z↦1/z then gives the full-plane statement. The proof adapts the method of traps of Calegari, Koch and Walker, using a neighboring-piece criterion, short chains through consecutive digits, and Rouché's theorem. We also classify the convex attractors.
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BibTeX
@misc{espigule2026regular,
title = {{Regular closedness off the real axis for root closures of bounded-height integer polynomials}},
author = {Bernat Espigule},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38971},
eprint = {2610.38971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38971},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regular closedness off the real axis for root closures of bounded-height integer polynomials
AU - Bernat Espigule
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38971
UR - https://arxiv.org/abs/2610.38971
AB - For each integer h≥1, let Z_h be the closure in the complex plane of the roots of nonzero integer polynomials whose coefficients have absolute value at most h, with no degree restriction. We prove that Z_h is regular closed off the real axis: every non-real point of Z_h is a limit of interior points. The root-closure identification of Espigulé and Juher, for which we include a self-contained proof, shows that the same closure arises from monic polynomials and that outside the closed unit disc it is the connectedness locus of h+1 equally spaced collinear similarities. The main step proves that this locus is regular closed off the real axis; this was known for h=1 by Calegari, Koch and Walker, for h≥19, and inside an explicit lens. The symmetry z↦1/z then gives the full-plane statement. The proof adapts the method of traps of Calegari, Koch and Walker, using a neighboring-piece criterion, short chains through consecutive digits, and Rouché's theorem. We also classify the convex attractors.
ER -