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Liang-Chung Hsia

Publications and source records attributed to Liang-Chung Hsia.

18 recordsLinked to original sources
arXiv2025

Zeta function and entropy for non-archimedean subhyperbolic dynamics

Let $K$ be a complete non-archimedean field of characteristic $0$ equipped with a discrete valuation. We establish the rationality of the Artin-Mazur zeta function on the Julia set for any subhyperbolic rational map defined over $K$ with a compact Julia set. Furthermore, we conclude that the topological entropy on the Julia set of such a map is given by the logarithm of a weak Perron number. Conversely, we construct a (sub)hyperbolic rational map defined over $K$ with compact Julia set whose topological entropy on the Julia set equals the logarithm of a given weak Perron number. This extends Thurston's work on the entropy for postcritically finite interval self-maps %of the unit interval to the non-archimedean setting.

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arXiv2023

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

The problem to construct optimal conflict-avoiding codes of even lengths and the Hamming weight $3$ is completely settled. On the contrary, it is still open for odd lengths. It turns out that the prime lengths are the fundamental cases needed to be constructed. In the article, we study conflict-avoiding codes of prime lengths and give a connection with the so-called cyclotomic numbers. By having some nonzero cyclotomic numbers, a well-known algorithm for constructing optimal conflict-avoiding codes will work for certain prime lengths. As a consequence, we are able to answer the size of optimal conflict-avoiding code for a new class of prime lengths.

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arXiv2023

Certain Diagonal Equations and Conflict-Avoiding Codes of Prime Lengths

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length $p$ and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form $g^2 X^{\ell} + g Y^{\ell} + 1 = 0$ over the finite field $\mathbb{F}_{p}$ for some primitive root $g$ modulo $p.$ We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field $\mathbb{F}_q$ of $\mathbb{F}_{p}.$ We show that for $q$ greater than a lower bound of the order of magnitude $O(\ell^2)$ there exists a generator $g$ of $\mathbb{F}_{q}^{\times}$ such that the equation in question is solvable over $\mathbb{F}_{q}.$ Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight $3.$

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arXiv2018

Heights and periodic points for one-parameter families of Hénon maps

In this paper we study arithmetic properties of a one-parameter family ${\mathbf H}$ of Hénon maps over the affine line. Given a family of initial points ${\mathbf P}$ satisfying a natural condition, we show the height function $h_{\mathbf P}$ associated to ${\mathbf H}$ and ${\mathbf P}$ is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We then show various local properties of $h_{\mathbf P}$. Next we consider the set $Σ({\mathbf P})$ consisting of periodic parameter values, and study when $Σ({\mathbf P})$ is an infinite set or not. We also study unlikely intersections of periodic parameter values.

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arXiv2018

Finite index theorems for iterated Galois groups of unicritical polynomials

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $β\in \overline{K}$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(β)=\mathop{\rm{Gal}}(K(f^{-n}(β))/K(β))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^\infty:G_\infty(β)]<\infty$ unless $β$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $\bigcup_{n=1}^\infty K(f_1^{-n}(β))$ and $\bigcup_{n=1}^\infty K(f_2^{-n}(β))$ are disjoint over a finite extension of $K$.

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arXiv2017

A variant of a theorem by Ailon-Rudnick for elliptic curves

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m_{1,t} and m_{2,t}, we have that [m_{i,t}](P_i)_t = (Q_i)_t on E_i (for i = 1,2), then at least one of the following conclusions must hold: either (i) there exists an isogeny f between E_1 and E_2 and also there exists a nontrivial endomorphism g of E_2 such that f(P_1) = g(P_2); or (ii) Q_i is a multiple of P_i for some i = 1,2. A special case of our result answers a conjecture made by Silverman.

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arXiv2016

Greatest common divisors of iterates of polynomials

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant compositionally independent polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $λ$ with the property that there is an $n$ such that $(x - λ)$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$.

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arXiv2016

Simultaneously preperiodic points for families of polynomials in normal form

Let $d>m>1$ be integers, let $c_1,\dots, c_{m+1}$ be distinct complex numbers, and let $\mathbf{f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ be an $m$-parameter family of polynomials. We prove that the set of $m$-tuples of parameters $(t_1,\dots, t_m)\in\mathbb{C}^m$ with the property that each $c_i$ (for $i=1,\dots, m+1$) is preperiodic under the action of the corresponding polynomial $\mathbf{f}(z)$ is contained in finitely many hypersurfaces of the parameter space $\mathbb{A}^m$.

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arXiv2015

Unlikely Intersection For Two-Parameter Families of Polynomials

Let $c_1, c_2, c_3$ be distinct complex numbers, and let $d\ge 3$ be an integer. We show that the set of all pairs $(a,b)\in \mathbb{C}\times \mathbb{C}$ such that each $c_i$ is preperiodic for the action of the polynomial $x^d+ax+b$ is not Zariski dense in the affine plane.

math.DS↗
arXiv2015

Ramification Filtrations of Certain Abelian Lie Extensions of Local Fields

Let $G\subset x{\mathbb F}_q[\![x]\!]$ ($q$ is a power of the prime $p$) be a subset of formal power series over a finite field such that it forms a compact abelian $p$-adic Lie group of dimension $d\ge 1$. We establish a necessary and sufficient condition for the APF extension of local field corresponding to $\left({\mathbb F}_q(\!(x)\!), G\right)$ under the field of norms functor to be an extension of $p$-adic fields. We then apply this result to study family of invertible power series with coefficients in a $p$-adic integers ring and commute with a fixed noninvertible power series under the composition of power series.

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arXiv2012

Preperiodic points for families of rational map

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.

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arXiv2012

Torsion points in families of Drinfeld modules

Let $Φ^ł$ be an algebraic family of Drinfeld modules defined over a field $K$ of characteristic $p$, and let $\bfa,\bfb\in K[ł]$. Assume that neither $\bfa(ł)$ nor $\bfb(ł)$ is a torsion point for $Φ^ł$ for all $ł$. If there exist infinitely many $ł\in\Kbar$ such that both $\bfa(ł)$ and $\bfb(ł)$ are torsion points for $Φ^ł$, then we show that for each $ł\in\Kbar$, we have that $\bfa(ł)$ is torsion for $Φ^ł$ if and only if $\bfb(ł)$ is torsion for $Φ^ł$. In the case $\bfa,\bfb\in K$, then we prove in addition that $\bfa$ and $\bfb$ must be $\Fpbar$-linearly dependent.

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arXiv2011

Preperiodic points for families of polynomials

Let $a(λ)$ and $b(λ)$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $λ$. We study whether there exist infinitely many complex numbers $λ$ such that both $a(λ)$ and $b(λ)$ are preperiodic for $f_λ$.

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arXiv2009

A quantitative estimate for quasi-integral points in orbits

Let f(z) be a rational function of degree at least 2 with coefficients in a number field K, and assume that the second iterate f^2(z) of f(z) is not a polynomial. The second author previously proved that for any b in K, the forward orbit O_f(b) contains only finitely many quasi-S-integral points. In this note we give an explicit upper bound for the number of such points.

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arXiv2008

On a Dynamical Brauer-Manin Obstruction

Let F : X --> X be a morphism of a variety defined over a number field K, let V be a K-subvariety of X, and let O_F(P)= {F^n(P) :n=0,1,2,...} be the orbit of a point P in X(K). We describe a local-global principle for the intersection of V and O_F(P). This principle may be viewed as a dynamical analog of the Brauer-Manin obstruction. We show that the rational points of V(K) are Brauer--Manin unobstructed for power maps on P^2 in two cases: (1) V is a translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key tool in the proofs is the classical Bang-Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.

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arXiv2004

Canonical Heights, Transfinite Diameters, and Polynomial Dynamics

Let phi(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating phi gives rise to a dynamical system and a corresponding canonical height function, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of phi over various completions of K, and we apply this formula to give a generalization of Bilu's equidistribution theorem for sequences of points whose canonical heights tend to zero.

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