Abstract:
This paper contains two parts. In the first part, we shall show that the result given in the Zoladek's example [1], which claims the existence of 11 small-amplitude limit cycles around a singular point in a particular cubic vector filed, is not correct. Mistakes made in the paper [1] leading to the erroneous conclusion have been identified. In fact, only 9 small-amplitude limit cycles can be obtained from this example after the mistakes are corrected, which agrees with the result obtained later by using the method of focus value computation [2]. In the second part, we present an example by perturbing a quadratic Hamiltonian system with cubic polynomials to obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. This demonstrates a good example in applying higher-order Melnikov functions to study bifurcation of limit cycles.

Abstract:
Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points of $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha) = \beta$. The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers $n \ge 1$ and $d \ge 2$, there are only finitely many isomorphism classes of directed graphs $G(f,K)$ as one ranges over all number fields $K$ of degree $n$ and polynomials $f(z) \in K[z]$ of degree $d$. In the case $(n,d) = (1,2)$, Poonen has given a complete classification of all directed graphs which may be realized as $G(f,\mathbb{Q})$ for some quadratic polynomial $f(z) \in \mathbb{Q}[z]$, under the assumption that $f$ does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case $(n,d) = (2,2)$. By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen's -- with a similar assumption on points of large period -- but over all quadratic extensions of $\mathbb{Q}$.

Abstract:
It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal$_{\bold Q}$-stable $5$-cycles, and show that there exist Gal$_{\bold Q}$-stable $N$-cycles for infinitely many $N$. Furthermore, we answer a question of Morton by showing that the genus~$14$ curve and its quotient are not modular. Finally, we mention some partial results for $N=6$.

Abstract:
In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. Based on the method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be elementary center or nilpotent center. Under the condition for the singular point to be a center, we obtain the normal form of the Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating from the center using the algorithm to compute the coefficients of Melnikov function. Finally, perturbing the symmetric hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is same to that of another center.

Abstract:
We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically integrable by the set of functions h_1(t), ..., h_n(t) where h_1(t) is any solution of an ordinary differential equation of order n and h_k(t) are differential polynomials in h_1(t), k = 2, ..., n. We describe a method of constructing this ordinary differential equation. We give a classification of symmetric quadratic dynamical systems and describe the maximal subgroup in GL(n, C) that acts on this systems. We apply our results to analysis of classical systems of Lotka-Volterra type and Darboux-Halphen system and their modern generalizations.

Abstract:
We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

Abstract:
Bounding the number of preperiodic points of quadratic polynomials with rational coefficients is one case of the Uniform Boundedness Conjecture in arithmetic dynamics. Here, we provide a general framework that may reduce finding periodic points of such polynomials over Galois extensions of $\mathbb{Q}$ to finding periodic points over the rationals. Furthermore, we present evidence that there are no such polynomials (up to linear conjugation) with periodic points of exact period 5 in quadratic fields by searching for points on an algebraic curve that classifies quadratic periodic points of exact period 5 and suggesting the application of the method of Chabauty and Coleman for further progress.

Abstract:
In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are only finitely many regular ternary triangular forms. A more general discussion of integral quadratic polynomials over a Dedekind domain inside a global field is also given.

Abstract:
An ordering for Laurent polynomials in the algebraic torus $(\mathbb C^*)^D$, inspired by the Cantero-Moral-Vel\'azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus $\mathbb T^D$. The Gauss-Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. Christoffel type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used for the finding of a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for nice Laurent polynomials which are analyzed from the perspective of Newton polytopes and tropical geometry. Discrete and continuous deformations of the measure lead to a Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials.