Uniform unlikely intersections for unicritical polynomials

Q: What does this research mean for the field?

For the family of unicritical polynomials f_t(z)=z^d+t, there exists a uniform constant C(d) that bounds the number of parameters t for which any two points a and b (with a^d ≠ b^d) are simultaneously preperiodic. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Hang Fu

doi:10.46298/epiga.2026.10958

September 9, 2026Épijournal de Géométrie AlgébriqueOpen Access

Uniform unlikely intersections for unicritical polynomials

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Key Points

To establish a uniform upper bound on the number of parameters for which two distinct complex points are simultaneously preperiodic under a family of unicritical polynomials.
Analyzed the one-parameter family of unicritical polynomials defined by f_t(z) = z^d + t over the complex plane for degree d ≥ 2.
Evaluated the parameter set t where two distinct points a and b satisfy a^d ≠ b^d and both exhibit preperiodic dynamic orbits.
Proved the existence of a uniform constant C(d), dependent solely on the polynomial degree d, that bounds the number of shared preperiodic parameters.
Showed that the bound holds independently of the specific initial values of points a and b, provided their d-th powers are not equal.

Implication

Theoretical analysis reveals a uniform bound on simultaneous preperiodic parameters for unicritical polynomials, highlighting rigid constraints on unlikely intersections.

Abstract

Fix d≥2, and consider the family fₜ(z)=zᵈ+t of polynomials parameterized by t. In this article, we will show that there exists a constant $C(d)$ such that for any a,b with aᵈ≠ bᵈ, the number of t such that a and b are both preperiodic for fₜ is at most $C(d)$.

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Cite This Study

Hang Fu (2026) studied this question.

synapsesocial.com/papers/6aa27a0b58559d80afc72adb https://doi.org/10.46298/epiga.2026.10958

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Uniform unlikely intersections for unicritical polynomials

Key Points

Implication

Abstract

Discussion

Cite This Study

Uniform unlikely intersections for unicritical polynomials

Key Points

Implication

Abstract

Discussion

Cite This Study

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