三次多项式与有理函数的新范式

Misiurewicz polynomials for rational maps with nontrivial automorphisms

In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which the corresponding map is post-critically finite with a certain dynamical portrait. Then we prove that the polynomial is irreducible in certain cases.

A Geometric Approach to Polynomial and Rational Approximation

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.

Expressing Reachability in Linear Recurrences, as Infinite Determinants and Rational Polynomial Equations

We present two tools, which could be useful in determining whether or not a non-Homogenous Linear Recurrence can reach a desired rational. First, we derive the determinant that is equal to the ith term in a non-Homogenous Linear Recurrence. We use this to derive the infinite determinant that is zero, if and only if, the desired rational can be reached by some term in the recurrence. Second, we derive an infinite summation of rational Polynomials, such that this summation can be equal to 1, if and only if, the desired rational can be reached by some term in the recurrence.

一个具有有理系数且无局部多项式李雅普诺夫函数的全局渐近稳定的多项式向量场

A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

本文提出一个二维七次多项式向量场，具有有理系数，全局渐近稳定但无局部解析李雅普诺夫函数。

The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we show under this assumption that the number of preperiodic points is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular curves $X_1(13)$, $X_1(16)$, and $X_1(18)$ all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a non-mod…

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.

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 …

On a characterization of polynomials among rational functions in non-archimedean dynamics

We study a question on characterizing polynomials among rational functions of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.

有理范式中小除数的精确全局控制

Exact Global Control of Small Divisors in Rational Normal Form

本文构建对小除数具有精确全局控制的有理范式，并用于证明 Gevrey 空间中非线性 Schrödinger 方程一般小初始数据的次指数长时间稳定性。

Families of polynomials of every degree with no rational preperiodic points

Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$.

Completely invariant sets of normality for rational semigroups

Let G be a semigroup of rational functions of degree at least two where the semigroup operation is composition of functions. We prove that the largest open subset of the Riemann sphere on which the semigroup G is normal and is completely invariant under each element of G, can have only 0, 1, 2, or infinitely many components.