For every integer d≥ 2 and every number field K, there exists a constant epsilon_K,d>0 such that alpha∈operatornameMPoly^d(K^ab):h_crit(alpha)<epsilon_K,d=alpha∈operatornameMPoly^d(K^ab):h_crit(alpha)=0 is finite, where h_crit denotes the critical height on the moduli space of degree-d polynomials.

Bogomolov property and finiteness of postcritically finite polynomials over abelian extensions

For every integer $d\ge 2$ and every number field $K$ , there exists a constant $\epsilon_{K,d}>0$ such that $\left\{\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}}):h_{\mathrm{crit}}(\alpha)<\epsilon_{K,d}\right\}=\left\{\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}}):h_{\mathrm{crit}}(\alpha)=0\right\}$ is finite, where $h_{\mathrm{crit}}$ denotes the critical height on the moduli space of degree- $d$ polynomials.

References

Primary source

arXiv

Additional references

A Bogomolov property for moduli spaces of polynomials over abelian extensions — arXiv — Geng-Rui Zhang

arithmetic-dynamics diophantine-geometry

Posted 2 days ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed solved

An unrefereed preprint claims to prove the finiteness theorem, but the claim has not been independently checked.

The problem asks whether polynomial moduli over abelian extensions have the Bogomolov property, implying finiteness of postcritically finite polynomials. No earlier proposer or dated classical result is identified in the retrieved material.

October 5, 2026 preprint report

Geng-Rui Zhang's preprint A Bogomolov property for moduli spaces of polynomials over abelian extensions claims the result using adelic line bundles, ramification estimates, and a correspondence argument. If correct, this establishes the advertised height and finiteness theorem; the claim is unrefereed and independently uncorroborated.

Current status (as of October 2026): A preprint claims the theorem, but no independent verification is recorded, so its resolution remains unconfirmed.

Sources

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