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Wade Hindes

Publications and source records attributed to Wade Hindes.

At least 19 recordsLinked to original sources
arXiv2025

Preperiodic integers for $x^d+c$ in large degree

Given a number field $K$, we completely classify the preperiodic portraits of the maps $x^d+c$ where $c\in K$ is an algebraic integer and $d$ is sufficiently large depending on the degree of $K$. Specifically, we show that there are exactly thirteen such portraits up to the natural action of roots of unity. In particular, we obtain some of the main results of recent work of the authors unconditionally for algebraic integers by replacing the use of the abc-conjecture with bounds on linear forms in logarithms. We then include applications of this work to several problems in semigroup dynamics, including the construction of irreducible polynomials and the classification of post-critically finite sets.

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arXiv2025

Prime-powered images and irreducible polynomials in dynamical semigroups

Let $G=\langle x^d+c_1,\dots,x^d+c_s\rangle$ be a semigroup generated under composition for some $c_1,\dots,c_s\in\mathbb{Z}$ and some $d\geq2$. Then we prove that, outside of an exceptional one-parameter family, $G$ contains a large and explicit subset of irreducible polynomials if and only if it contains at least one irreducible polynomial. In particular, this conclusion holds when $G$ is generated by at least $s\geq3$ polynomials when $d$ is odd and at least $s\geq5$ polynomials when $d$ is even. To do this, we prove a classification result for prime powered iterates under $f(x)=x^d+c$ when $c\in\mathbb{Z}$ is nonzero. Namely, if $f^n(\alpha)=y^p$ for some $n\geq4$, some $\alpha,y\in\mathbb{Z}$, and some prime $p|d$, then $\alpha$ and $y^p$ are necessarily preperiodic and periodic points for $f$ respectively. Moreover, we note that $n=4$ is the smallest possible iterate for which one may make this conclusion.

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arXiv2025

On the factorization of iterates of $x^d+c$ in large degree

Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\alpha\in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\geq2$ and some $c\in K$. Then for many $c$ and $d$, we prove that $f_{d,c}^n(x)-\alpha$ has at most $d$ factors in $K[x]$ for all $n\geq1$. For example, when $\alpha=0$ we prove that the set \[\Big\{d\,:\, f_{d,c}^n(x)\;\text{has at most $d$ factors in $K[x]$ for all $n\geq1$ and all $h(c)>0$}\Big\}\] has positive asymptotic density. We then apply this result to compute the density of prime divisors in certain forward orbits and to establish the finiteness of integral points in certain backward orbits.

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arXiv2024

Unicritical polynomials over $abc$-fields: from uniform boundedness to dynamical Galois groups

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely classify all portraits of $K$-rational preperiodic points for such $\phi$ for all sufficiently large degrees $d$. More precisely, we prove that, up to accounting for the natural action of $d$th roots of unity on the preperiodic points for $\phi$, there are exactly thirteen such portraits up to isomorphism. In particular, for all such global fields $K$, it follows from our results together with earlier work of Doyle-Poonen and Looper that the number of $K$-rational preperiodic points for $\phi$ is uniformly bounded -- independent of $d$. That is, there is a constant $B(K)$ depending only on $K$ such that \[\big|\text{PrePer}(x^d+c,K)\big|\leq B(K)\] for all $d\geq2$ and all $c\in K$. Moreover, we apply this work to construct many irreducible polynomials with large dynamical Galois groups in semigroups generated by sets of unicritical polynomials under composition.

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arXiv2023

On the proportion of irreducible polynomials in unicritically generated semigroups

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. Then we construct a large, explicit subset of irreducible polynomials within the semigroup generated by $S$ under composition; in fact, we show that this subset has positive asymptotic density within the full semigroup when we count polynomials by degree. In addition, when $p=2$ or $3$ we construct an infinite family of semigroups that break the local-global principle for irreducibility. To do this, we use a mix of algebraic and arithmetic techniques and results, including Runge's method, the elliptic curve Chabauty method, and Fermat's Last Theorem.

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arXiv2023

Counting points by height in semigroup orbits

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.

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arXiv2023

The size of semigroup orbits modulo primes

Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{p}$ of $K$, let $m_{\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\bmod\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\operatorname{\mathsf{N}}_{K/\mathbb{Q}}\mathfrak{p}$ is a set of density zero. For $V=\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.

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arXiv2023

Galois groups and prime divisors in random quadratic sequences

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $γ$ of elements of $S$, one can associate an arboreal representation to $γ$, generalizing the case of iterating a single polynomial. We study the probability that a random sequence $γ$ produces a ``large-image'' representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets $S$ defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all $S$ possessing a particular kind of obstruction that generalizes the post-critically finite case in single-polynomial iteration.

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arXiv2022

Dynamical Diophantine Approximation Exponents in Characteristic $p$

Let $ϕ(z)$ be a non-isotrivial rational function in one-variable with coefficients in $\overline{\mathbb{F}}_p(t)$ and assume that $γ\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ is not a post-critical point for $ϕ$. Then we prove that the diophantine approximation exponent of elements of $ϕ^{-m}(γ)$ are eventually bounded above by $\lceil d^m/2\rceil+1$. To do this, we mix diophantine techniques in characteristic $p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point $a\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ and write $ϕ^n(a)=a_n/b_n$ for some coprime polynomials $a_n,b_n\in\overline{\mathbb{F}}_p[t]$, then we prove that \[ \frac{1}{2}\leq \liminf_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)} \leq\limsup_{n\rightarrow\infty} \frac{\text{deg}(a_n)}{\text{deg}(b_n)}\leq2,\] whenever $0$ and $\infty$ are both not post-critical points for $ϕ$. In characteristic $p$, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.

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arXiv2021

Integral points in orbits in characteristic $p$

We prove a characteristic $p$ version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.

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arXiv2020

Orbit counting in polarized dynamical systems

We extend recent orbit counts for finitely generated semigroups acting on $\mathbb{P}^N$ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.

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arXiv2020

Counting points of bounded height in monoid orbits

Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.

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arXiv2020

Eventually stable quadratic polynomials over $\mathbb{Q}$

We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$ \textit{eventually stable} (over $\mathbb{Q}$). Previous work of Hamblen, Jones, and Madhu shows that $f_r$ is eventually stable unless $r$ has the form $1/c$ for some integer $c \not\in \{0,-1\}$, in which case existing methods break down. We study this family, and prove that several conditions on $c$ of various flavors imply that all iterates of $f_{1/c}$ are irreducible. We give an algorithm that checks the latter property for all $c$ up to a large bound $B$ in time polynomial in $\log B$. We find all $c$-values for which the third iterate of $f_{1/c}$ has at least four irreducible factors, and all $c$-values such that $f_{1/c}$ is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of $f_{1/c}$, for all $c$ with absolute value at most $10^9$.

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arXiv2020

Dynamical height growth: left, right, and total orbits

Let $S$ be a set of dominant rational self-maps on $\mathbb{P}^N$. We study the arithmetic and dynamical degrees of infinite sequences of $S$ obtained by sequentially composing elements of $S$ on the right and left. We then apply this insight to dynamical Galois theory.

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arXiv2018

Finite orbit points for sets of quadratic polynomials

Let $S=\{x^2+c_1, x^2+c_2,\dots, x^2+c_s\}$ be a set of quadratic polynomials with rational coefficients, and let $P$ be a rational basepoint. We classify the pairs $(S,P)$ for which $P$ has finite orbit for $S$, assuming that the maximum period length for each individual polynomial is at most three (conjectured by Poonen). In particular, under these hypotheses we prove that if $s\geq4$, then there are no points $P$ with finite orbit for $S$. Moreover, we use this perspective to formulate an analog of the Morton-Silverman Conjecture for sets of rational functions.

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arXiv2018

Stochastic Canonical Heights

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.

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