arXiv subjects

Wayne Peng

Publications and source records attributed to Wayne Peng.

10 recordsLinked to original sources

arXiv2025

Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

This paper investigates the dynamical properties of the Dickson polynomials $D_n(x, \alpha)$ of the first kind over finite fields, with an emphasis on the periodicity and algebraic structure of their iterated sequences. We consider the sequence $[D_n(x, \alpha) \pmod{x^q - x}]_{n\geq1}$, and determine the exact period of this sequence. We then use the classical functional equation for the Dickson polynomials to relate the dynamics of $D_n(x,\alpha)$ to the power map $u\mapsto u^n$ on a suitable subset of $\mathbb F_{q^2}^\times$. In the permutation case $\gcd(n,q^2-1)=1$, this gives an explicit description of the group of Dickson polynomial functions under composition. We also obtain partial structural result in the non-permutation case. As further applications, we derive several new identities for the Dickson polynomials. Finally, we identify and prove a symmetry property of the Dickson polynomial family.

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arXiv2025

Overgroups of the arboreal representation of PCF polynomial

Consider a number field $K$ and a rational function $f$ of degree greater than 1 over $K$. By taking preimages of $\alpha\in K$ under successive iterates of $f$, an infinite $d$-ary tree $T_\infty$ rooted at $\alpha$ can be constructed. An edge is assigned between two preimages $x$ and $y$ if $f(x)=y$. The absolute Galois group of $K$, acting on $T_\infty$ through tree automorphisms, generates a subgroup $\text{Gal}_f^\infty(\alpha)$ in the group of all automorphisms of $T_\infty$, $\text{Aut}(T_\infty)$. We have discovered a new class of natural overgroups in which the image of the Galois representation attached to a PCF polynomial must reside. Moreover, we have found that the image of the Galois representation of a new PCF polynomial is isomorphic to one of these overgroups. We also investigate the structure of these overgroups for specific maps, such as normalized dynamical Belyi polynomials, and show that the normal subgroups of these overgroups form a unique chief series. This allows us to bound the number of generators through group-theoretic analysis.

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arXiv2022

Integrality and Thurston Rigidity for Bicritical PCF Polynomials

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.

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arXiv2021

A Tits alternative for rational functions

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field that do not have the same set of preperiodic points, then the semigroup generated by f and g contains a nonabelian free semigroup.

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arXiv2020

A Unique Chief Series in the arboreal Galois Group of Belyi Maps

We give a complete description of the normal subgroups of arboreal Galois groups of Belyi maps. The normal groups form a unique chief series. We also carefully compute the discriminate of the iterate of a polynomial minus an algebraic number, which allows us to predict when a such discriminate is a perfect square in the base field or intermediate field for a postcritically finite polynomials (PCF). As a consequence we are able to find another PCF cubic polynomial that has the same arboreal Galois group as the one of Belyi maps.

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arXiv2020

Unlikely intersection problems for restricted lifts of p-th power

I define a morphism on $\mathbb{C}_p$ called a lift of $p$-th power if its natural restriction to the residue field of $\mathbb{C}_p$ is a $p$-th power of some morphism. This definition generalizes from the lift of Frobenius. In this paper the theory of perfectoid spaces is applied on the morphism to the Manin-Mumford conjecutre, the Mordell-Lang conjecture and the Tate-Volch conjecture.

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arXiv2020

New Normal Forms For Degree Three Polynomials and Rational Functions

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared to rationality of the defining polynomials of the representation. We give a normal form for degree three polynomials which has the property that the set of fixed points is equal to the set of fixed point multipliers. This normal form is given in terms of moduli space invariants and, hence, has nice rationality properties. We further classify all degree three rational maps which can be conjugated to have a similar relationship between the fixed points and the fixed point multipliers.

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arXiv2019

Permutation polynomials: iteration of shift and inversion maps over finite fields

We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group generated by shift and inversion maps is used to study the randomness properties of permutation polynomials.

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arXiv2015

ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields

In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterl\'{e}-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.

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arXiv2015

Wall's Conjecture and the ABC Conjecutre

We show that the $abc$ conjecture of Masser-Oesterl\'{e}-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.

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