Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity
Forum of Mathematics, Sigma, Tome 12 (2024)
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In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$, where p is a prime number and where the orbit of $0$ is finite. For example, if $p=2$ and $0$ is periodic under $T^2+c$ with $c\in \mathbb {R}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in $\mathbb {Q}(y)[T]$.
@article{10_1017_fms_2023_112,
author = {P. Habegger and H. Schmidt},
title = {Lower {Bounds} for the {Canonical} {Height} of a {Unicritical} {Polynomial} and {Capacity}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {12},
year = {2024},
doi = {10.1017/fms.2023.112},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.112/}
}
TY - JOUR AU - P. Habegger AU - H. Schmidt TI - Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity JO - Forum of Mathematics, Sigma PY - 2024 VL - 12 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.112/ DO - 10.1017/fms.2023.112 LA - en ID - 10_1017_fms_2023_112 ER -
%0 Journal Article %A P. Habegger %A H. Schmidt %T Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity %J Forum of Mathematics, Sigma %D 2024 %V 12 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.112/ %R 10.1017/fms.2023.112 %G en %F 10_1017_fms_2023_112
P. Habegger; H. Schmidt. Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2023.112
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