Unconditional uniform boundedness of preperiodic points for unicritical polynomials $z^d+c$ over number fields

Nicole R. Looper (Inventiones mathematicae 225 (2021) 1–44) proved that for every degree $d\ge2$ and number field $K$ there is a constant $B(d,K)$ bounding the number of $K$ -rational preperiodic points of every unicritical polynomial $f(z)=z^d+c$ with $c\in K$ — but the proof assumes the classical $abc$ -conjecture for $d\ge5$ and the stronger, still-open $abcd$ -conjecture for $2\le d\le4$ . This problem asks for an unconditional proof of that uniform bound over number fields, or a refutation exhibiting a fixed $(d,K)$ with unboundedly many preperiodic points as $c$ varies. Even the flagship case $(d,K)=(2,\mathbb{Q})$ is open: as of 2026 even the weaker periodic-points form of the Morton–Silverman uniform boundedness conjecture is, in Rajagopal–Zhang's words, 'not known unconditionally for any tuple $(N,d,D)$ , even when $(N,d,D)=(1,2,1)$ '.

Background

Arithmetic dynamics studies the behavior of points under repeated application of a rational map. A point is called preperiodic if its forward orbit is eventually periodic, i.e. $f^{m+n}(P)=f^m(P)$ for some $m\ge0$ , $n\ge1$ ; for maps over number fields these are exactly the points of canonical height zero, the dynamical analogue of torsion points on abelian varieties. The Uniform Boundedness Conjecture of Morton and Silverman (1994) predicts that for all integers $D,N\ge1$ and $d\ge2$ there is a constant $B(D,N,d)$ such that every morphism $\varphi:\mathbb{P}^N\to\mathbb{P}^N$ of degree $d$ defined over a number field $K$ of degree $D$ has at most $B(D,N,d)$ preperiodic points in $\mathbb{P}^N(K)$ . It is a vast dynamical generalization of the torsion-boundedness theorems of Mazur and Merel for elliptic curves, and it remains open in essentially every instance. Its simplest nontrivial test family consists of the unicritical polynomials — polynomials with a single finite critical point — which, up to conjugacy, are exactly the maps $f_{d,c}(z)=z^d+c$ with $c\in K$ . Here the conjecture reduces to: for each $d\ge2$ and number field $K$ there is $B(d,K)$ with $\#\mathrm{PrePer}(z^d+c,K)\le B(d,K)$ for all $c\in K$ . Poonen (1998) showed for $K=\mathbb{Q}$ , $d=2$ that if $f_{2,c}$ has no rational periodic point of period 4 or higher, then it has at most 9 rational preperiodic points, and gave a refined conjectural classification; since periods 4 and 5 were excluded unconditionally by Morton and by Flynn–Poonen–Schaefer, respectively, the hypothesis is equivalent to excluding periods at least 6. Benedetto (2007) proved general bounds of order $O(s\log s)$ — concretely $\#\mathrm{PrePer}(f,K)\le C(d)\,s\log s$ — where $s$ is the number of places of bad reduction of $f$ (non-archimedean primes at which $f$ does not reduce to a polynomial map of the same degree; for $z^d+c$ , the primes dividing a denominator of $c$ ). The central obstacle is that $s$ is unbounded as $c$ varies, so all such bounds fail to uniformize over the family. Looper's theorem (Theorem 1.2 of the cited paper) overcomes this obstacle conditionally: for $K$ a number field (or a one-variable function field of characteristic zero) and $f(z)=z^d+c$ non-isotrivial (isotriviality — being conjugate over $\overline{K}$ to a map defined over the constant subfield — being a condition that can arise only over function fields), such a $B(d,K)$ exists, assuming the $abc$ -conjecture for $K$ when $d\ge5$ and the $abcd$ -conjecture when $2\le d\le4$ . The $abc$ -conjecture over a number field asserts, roughly, that for coprime nonzero $a,b,c\in K$ with $a+b=c$ , the logarithmic height of the triple is at most $(1+\epsilon)$ times its radical (the height contribution of the primes dividing $abc$ ) plus a constant. The $abcd$ -conjecture (Conjecture 2.1 of the cited paper; the $n=3$ case is $abc$ ) asserts: for $n\ge3$ , $\epsilon>0$ , there is a proper Zariski-closed subset $\mathcal{Z}$ of the hyperplane $\mathcal{H}:Z_1+\cdots+Z_n=0$ in $\mathbb{P}^{n-1}$ and a constant $C$ such that every point $P=(z_1,\dots,z_n)\in\mathcal{H}(K)\setminus\mathcal{Z}$ with all $z_i\ne0$ satisfies $h(P)<(1+\epsilon)\,\mathrm{rad}(P)+C$ , where $h$ is the standard logarithmic Weil height and $\mathrm{rad}(P)$ its radical part. It is a special case of Vojta's conjecture with truncated counting function (a higher-dimensional generalization of $abc$ for divisors with normal crossings), and it is open over number fields; the remark following Conjecture 2.1 notes that for the purposes of proving Theorem 1.2 the coefficient $1+\epsilon$ 'cannot simply be replaced by a larger constant', so weaker known radical–height bounds do not substitute for the dynamical application. The proof strategy produces special algebraic configurations (hexagons for $d=2$ , quadrilaterals for $d=3,4$ , triangles for $d\ge5$ ) whose vertices are $K$ -rational preperiodic points and whose coordinate differences would violate the assumed inequality; local ( $p$ -adic) equidistribution of preperiodic points on the $v$ -adic Julia sets (the nonarchimedean counterpart of the classical Julia set of complex dynamics) is transposed into global adelic control. For function fields, Looper's own introduction notes the theorem is superseded by the unconditional work of Doyle–Poonen, so the unresolved content of Theorem 1.2 is over number fields: whether the uniform bound exists with no unproved Diophantine hypothesis at all.

Problem Statement

Prove, unconditionally — that is, without assuming the $abc$ -conjecture, the $abcd$ -conjecture (Conjecture 2.1 of the cited paper), Vojta's conjecture, or any other unproved Diophantine hypothesis — that for every number field $K$ and every integer $d\ge2$ there exists a constant $B(d,K)$ such that for every $c\in K$ the unicritical polynomial $f(z)=z^d+c$ has at most $B(d,K)$ preperiodic points in $K$ (a preperiodic point is a point of $K$ whose forward orbit under iteration of $f$ is eventually periodic). Equivalently: remove from Looper's Theorem 1.2 the $abc$ -assumption used for $d\ge5$ and the $abcd$ -assumption used for $2\le d\le4$ over number fields. The flagship open special case is $(d,K)=(2,\mathbb{Q})$ : prove there exists $B$ with $\#\mathrm{PrePer}(z^2+c,\mathbb{Q})\le B$ for all $c\in\mathbb{Q}$ . A refutation is equally admissible as an answer: exhibit a fixed degree $d\ge2$ and number field $K$ for which $\#\mathrm{PrePer}(z^d+c,K)$ is unbounded as $c$ ranges over $K$ .

References

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