Solution of the Center Problem for a Class of Polynomial Differential Systems

Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.

The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3

Most studies of the time-reversibility are limited to a linear or an affine involution.In this paper,the authors consider the case of a quadratic involution.For a polynomial differential system with a linear part in the standard form(-y,x)in R~2,by using the method of regular chains in a computer algebraic system,the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given.When the system is quadratic,the necessary and sufficient conditions can be completely obtained by this procedure.For some cubic systems,the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained.These conditions can guarantee the corresponding systems to have a center.Meanwhile,a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution,then its phase diagram is symmetric about a parabola.

POLYNOMIAL DIFFERENTIAL SYSTEM WITH DEGENERATE INFINITY

In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.

Zeros of Differential Polynomial f（z）f（k）（z） -- a and Its Normality

Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f（z）f（k） （z）.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

Value Distribution of a Class of Differential Polynomials

Let f（z） be a transcendental meromorphic function in the complex plane and a ≠0 be a constant, for any positive integer m, n, k, satisfy m ≥ nk＋n＋2, ψ= f^m ＋a（f^（κ））^n has infinitely many zeros. The corresponding normal criterion also is proved.

Value Distribution of Differential Polynomials in Meromorphic Functions

Let f（z） be a meromorphic function and ψ be the differential polynomial of f which satisfies the condition of -↑N（r, f）＋-↑N （r, 1/f） = S（r, f）. We obtain several results about the zero point of the ψ and those results extend and improve the results of Yang and Yi in this paper.

Shared Values of Certain Nonlinear Differential Polynomials of Meromorphic Functions

In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.

The Existence of Periodic Solutions of a Class of <i>n</i>-Degree Polynomial Differential Equations*

This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.

Notes on Zeros of Differential Polynomials

The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.

DIFFERENTIAL POLYNOMIALS SHARING ONE VALUE

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and g^n＋ ag^（k）share b CM and the b-points of f^n＋ af^（k）are not the zeros of f and g, then f and g are either equal or closely related.

ON VALUE DISTRIBUTIONS OF GENERAL DIFFERENTIAL MONOMIALS

Value distributions of the general differential monomials is discussed.The following theorem is obtained:Let f be a transcendental meromorphic function in the plane,F=f n 0 (f (i) ) n i …(f (k) ) n k -c,n i≥1,c≠0 be a constant then (n 0-2)T(r,f)≤(r,1F)+S(r,f) when n 0】2;T(r,f)≤7(i+1)i( i) (r,1f)+(r,1F))+S(r,f) when n 0=1;T(r,f)≤7(N(r,1f)+(r,1F))+S(r,f) when n 0=0.

SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS

Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +

NORMALITY AND THE NUMBER OF ZEROS

In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.

INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIALDIFFERENTIAL SYSTEMS

We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.

A Triangular Decomposition Algorithm for Differential Polynomial Systems with Elementary Computation Complexity

In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of polynomials in one step using the theory of multivariate resultant. This seems to be the first differential triangular decomposition algorithm with elementary computation complexity.

On Miranda's Normal Criterion

In this paper, we study the normality of a family of analytic functions and prove the following theorem. Let F be a family of analytic functions in a domain D , k be a positive integer and a(z) , a 1(z) , a 2(z) , ... , a k(z) be analytic in D such that a(z)0 . If f(z)≠0 and the zeros of f (k) (z)+a 1(z)f (k-1) (z)+...+a k(z)f(z)-a(z) are of multiplicity at least 2 for each f∈F , then F is normal in D . This result improves Miranda s norm...

On Homogeneous Differential Polynomials of Meromorphic Functions

In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ∈ N. Moreover, an analogous normality criterion isobtained.

NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.