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Trevor Hyde

Publications and source records attributed to Trevor Hyde.

At least 19 recordsLinked to original sources
arXiv2026

The Arithmetic of Semirings Part I: Ideals

We study ideals in the semiring $\mathbb{N}$ of natural numbers, with a focus on those which are lost when extending from $\mathbb{N}$ to $\mathbb{Z}$. This leads to a new perspective on the classical theory of numerical semigroups, including the introduction of a natural multiplicative structure. We prove that unique factorization of ideals fails in $\mathbb{N}$ on several levels, introduce a handful of new tropical multiplicative invariants of numerical semigroups, characterize integral closures of ideals in terms of Newton polygons, and analyze the behavior of classical numerical semigroup invariants with respect to the product operation.

math.AC↗
arXiv2026

Rational orbits under correspondences

Consider an algebraic function like $F(x) = \sqrt{x^3 - 1}$. If $p \in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of correspondences on curves and their iterates. In this paper we show that if $F$ is a correspondence from $\mathbb{P}^1$ to itself defined over a finitely generated field $K$ of characteristic 0 satisfying several minor constraints, then either for each $n \geq 12$ there are only finitely many $p \in \mathbb{Q}$ for which $F^n(p)$ contains a $K$-rational point or $F$ belongs to an explicit list of known exceptional correspondences.

math.NT↗
arXiv2025

Profinite Iterated Monodromy Groups of Unicritical Polynomials

Let $f(x) = ax^d + b \in K[x]$ be a unicritical polynomial with degree $d \geq 2$ which is coprime to $\mathrm{char} K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this group, and use this analysis to determine the constant field extension in $K(f^{-\infty}(t))/K(t)$.

math.NT↗
arXiv2022

Polynomials with many rational preperiodic points

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d) \leq d$ such that $f_d(x)$ has at least $d + \lfloor \log_2(d)\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many integers $d$, the polynomials $f_d(x)$ and $f_d(x) + 1$ have at least $d^2 + d\lfloor \log_2(d)\rfloor - 2d + 1$ common complex preperiodic points.

math.DS↗
arXiv2021

Dynamical moduli spaces and polynomial endomorphisms of configurations

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.

math.AG↗
arXiv2021

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

Given a generic polynomial $f(x)$, the generalized dynatomic polynomial $Φ_{f,c,d}(x)$ vanishes at precisely those $α$ such that $f^c(α)$ has period exactly $d$ under iteration of $f(x)$. We show that the shifted dynatomic polynomials $Φ_{f,c,d}(x) - 1$ often have generalized dynatomic factors, and that these factors are in correspondence with certain cyclotomic factors of necklace polynomials. These dynatomic factors of $Φ_{f,c,d}(x) - 1$ have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x)$.

math.NT↗
arXiv2021

Density of Periodic Points for Lattès maps over Finite Fields

Let $L_d$ be the Lattès map associated to the multiplication-by-$d$ endomorphism of an elliptic curve $E$ defined over a finite field $\mathbb{F}_q$. We determine the density $δ(L_d,q)$ of periodic points for $L_d$ in $\mathbb{P}^1(\mathbb{F}_q)$. We show that the periodic point densities $δ(L_d,q^n)$ converge as $n \rightarrow \infty$ along certain arithmetic progressions, and compute simple explicit formulas for $δ(L_\ell,q)$ when $\ell$ is a prime and $E$ belongs to a special family of supersingular elliptic curves.

math.NT↗
arXiv2021

Cyclotomic factors of necklace polynomials

We observe that the necklace polynomials $M_d(x) = \frac{1}{d}\sum_{e\mid d}μ(e)x^{d/e}$ are highly reducible over $\mathbb{Q}$ with many cyclotomic factors. Furthermore, the sequence $Φ_d(x) - 1$ of shifted cyclotomic polynomials exhibits a qualitatively similar phenomenon, and it is often the case that $M_d(x)$ and $Φ_d(x) - 1$ have many common cyclotomic factors. We explain these cyclotomic factors of $M_d(x)$ and $Φ_d(x) - 1$ in terms of what we call the \emph{$d$th necklace operator}. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.

math.CO↗
arXiv2018

Liminal reciprocity and factorization statistics

Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined rational function $M_{d,\infty}(q)$. Furthermore we show that the limit $M_{d,\infty}(q)$ is related to the classic necklace polynomial $M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity.

math.NT↗
arXiv2018

Normal elements in finite fields

We give a simple derivation of the formula for the number of normal elements in an extension of finite fields. Our proof is based on the fact that units in the Galois group ring of a field extension act simply transitively on normal elements.

math.NT↗
arXiv2018

Factorization statistics and the twisted Grothendieck-Lefschetz formula

We announce recent results on a connection between factorization statistics of polynomials over a finite field and the structure of the cohomology of configurations in $\mathbb{R}^3$ as a representation of the symmetric group. This connection parallels a result of Church, Ellenberg, and Farb relating factorization statistics of squarefree polynomials and the cohomology of configurations in $\mathbb{R}^2$.

math.NT↗
arXiv2018

Polynomial factorization statistics and point configurations in $\mathbb{R}^3$

We use generating functions to relate the expected values of polynomial factorization statistics over $\mathbb{F}_q$ to the cohomology of ordered configurations in $\mathbb{R}^3$ as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.

math.RT↗
arXiv2017

Collusions in Teichmüller expansions

If $\mathfrak{p} \subseteq \mathbb{Z}[ζ]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $\mathbb{Z}$, then every element $α$ of the completion $\mathbb{Z}[ζ]_\mathfrak{p}$ has a unique expansion as a power series in $p$ with coefficients in $μ_{p^d -1} \cup \{0\}$ called the Teichmüller expansion of $α$ at $\mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmüller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $\mathbb{Z}[ζ]$.

math.NT↗
arXiv2017

Circulant q-Butson Hadamard matrices

If $q = p^n$ is a prime power, then a $d$-dimensional \emph{$q$-Butson Hadamard matrix} $H$ is a $d\times d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d = p^m$ and then explicitly construct a family of examples in all possible dimensions. These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.

math.CO↗
arXiv2017

Polynomial splitting measures and cohomology of the pure braid group

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $λ$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)

math.RT↗
arXiv2015

Gauss' hidden menagerie: from cyclotomy to supercharacters

Gaussian periods, when viewed appropriately, exhibit a dazzling and eclectic host of visual qualities. This brief survey reviews the historical context and summarizes our current knowledge of graphical properties of Gaussian periods.

math.NT↗
arXiv2015

Small dynamical heights for quadratic polynomials and rational functions

Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height $\hat{h}_f(x)$ of a non-preperiodic rational point $x$ is bounded below by a uniform multiple of the height of $f$ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range of previous searches.

math.NT↗