arXiv subjects

Geng-Rui Zhang

Publications and source records attributed to Geng-Rui Zhang.

9 recordsLinked to original sources

arXiv2026

Generic reconstruction of rational maps from multipliers of periods one and two

For every integer $d\geq2$ and every field of characteristic different from $2$, we prove that on the moduli space $\mathcal{M}_d$ of degree-$d$ rational maps, the multiplier spectrum morphism formed from the periodic points of periods one and two is birational to the closure of its image. Consequently, over every algebraically closed field of characteristic different from $2$, it is generically injective, which proves a recent conjecture of Ji and Xie in characteristic zero. The proof uses a fixed-index normal form, a non-archimedean degeneration of two-cycles, and birational reconstruction of an affine fixed-point configuration from pair invariants.

math.DS↗

arXiv2026

$13$ unknowns over quadratic integer rings and Lucas congruences

For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\mathcal{O}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \[ P(Y_1,\ldots,Y_{13})\in\mathbb{Z}[Y_1,\ldots,Y_{13}],\quad \mathrm{deg}\ P\leq D_0, \] decides whether $P=0$ has a solution in $\mathcal{O}_K^{13}$. The arithmetic input is a fourth-order Pell--Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas--Wieferich and Wall--Sun--Sun primes. We also establish local surjectivity and $\ell$-adic density for second-order correction terms for norm-one Lucas sequences.

math.NT↗

arXiv2026

Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \exists\lambda\in\mathbb{C}, f^{\circ m}(\lambda)=g^{\circ n}(\lambda)=c(\lambda)\right\rbrace\] is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case $m=n$. We also obtain partial results on recurrence sets for rational maps in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank $\leq2$.

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arXiv2026

Rigidity of Lyapunov exponents for polynomials

Let $f,g\in\overline{\mathbb{Q}}[z]$ be polynomials of degree $d\geq2$ with disconnected Julia sets. We prove that they have the same Lyapunov exponent $\mathcal{L}_f=\mathcal{L}_g$ if and only if either $f$ and $g$ are intertwined, or $f$ and $\overline{g}$ are intertwined. The analogous result for critical heights is also obtained. As an application, we provide a new proof of the theorem stating that the multiplier spectrum morphism on the moduli space of polynomials is generically injective.

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arXiv2025

On the multiplier spectrum of polynomials

We prove several results on the multiplier spectrum of polynomials. We provide a detailed proof of the theorem stating that the multiplier spectrum morphism is generically injective on the moduli space of polynomials. We obtain a description of the non-injective locus of the multiplier spectrum morphism for polynomials of degree $d\geq2$. Roughly speaking, we prove that, apart from isolated exceptions, polynomials with the same multiplier spectrum are intertwined. More precisely, we show that, up to iteration and isolated exceptions, the polynomials are either equivalent or related by Ritt moves. We also investigate the relationship between Ritt moves and multiplier spectra over arithmetic progressions.

math.DS↗

arXiv2025

Cyclotomic integral points for affine dynamics

Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic $f$-preperiodic points is Zariski-dense in $\mathbb{A}^N$, then some iterate $f^{\circ l}$ ($l\geq1$) is a quotient of a surjective algebraic group endomorphism $g:\mathbb{G}_m^N\to\mathbb{G}_m^N$, over $\overline{\mathbb{Q}}$. This result generalizes a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials to higher dimensions. In fact, we prove a much more general rigidity result for dominant endomorphisms $f$ on an affine variety $X$ defined over a number field, concerning "almost $f$-invariant" Zariski-dense subsets of cyclotomic integral points. We apply our results to backward orbits of regular endomorphisms on $\mathbb{A}^N$ of algebraic degree $d\geq2$, and to periodic points of automorphisms of H\'enon type on $\mathbb{A}^N$.

math.DS↗

arXiv2023

Space spanned by characteristic exponents

We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by all the (finite) characteristic exponents of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$.

math.DS↗

arXiv2023

Realizability of Some Combinatorial Sequences

A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is equal to the number of periodic points of $T$ in $X$ of (not necessarily exact) period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Ap\'ery numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schr\"oder numbers are not almost realizable.

math.NT↗

arXiv2021

$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns

In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\not\in\mathbb Z\iff \exists x_1\cdots\exists x_{n}[P(t,x_1,\ldots,x_{n})=0]$$ where variables range over $\mathbb Q$, equivalently $$t\in\mathbb Z\iff \forall x_1\cdots\forall x_{n}[P(t,x_1,\ldots,x_{n})\not=0].$$ In this paper we prove that we may take $n=32$. Combining this with a result of Z.-W. Sun, we show that there is no algorithm to decide for any $f(x_1,\ldots,x_{41})\in\mathbb Z[x_1,\ldots,x_{41}]$ whether $$\forall x_1\cdots\forall x_9\exists y_1\cdots\exists y_{32}[f(x_1,\ldots,x_9,y_1,\ldots,y_{32})=0],$$ where variables range over $\mathbb Q$.

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