首页/论文/2602.23902

arXiv 2602.239022026-02-27math.CA

多项式与三角Abel方程的有理解与极限环

Rational solutions and limit cycles of polynomial and trigonometric Abel equations

Universitat de les Illes Ballears（巴利阿里群岛大学）

Luis Angel Calderon

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

AI总结：

本文研究多项式与三角Abel方程的有理解和极限环问题，探讨了多项式和三角多项式条件下解的数量界。

摘要：

我们研究Abel微分方程x₀ = A(t)x³ + B(t)x² + C(t)x。具体来说，我们确定当A(t)、B(t)和C(t)为实系数或复系数多项式时，其有理解的数量界；以及当A(t)、B(t)和C(t)为实系数三角多项式时，其有理极限环的数量界。

英文摘要：

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of rational limit cycles when A(t), B(t) and C(t) are trigonometric polynomials with real coefficients.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The integrals in Gradshteyn and Ryzhik. Part9: Combinations of logarithms, rational and trigonometric functions

The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals.

2007-07-14math.CA

Closed-form formulae for the derivatives of trigonometric functions at rational multiples of $π$

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $π$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.

2009-11-24math.CA

On generalized trigonometric functions and series of rational functions

Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.

2017-09-02math.CAmath.NT

幂级数、三角级数与切比雪夫多项式级数的一致有理逼近

Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials

本文研究三角级数与切比雪夫多项式级数的一致有理逼近，建立三角 Hermite-Padé 多项式的存在唯一性准则并给出显式形式，同时构造例子说明 Hermite-Jacobi 逼近与 Hermite-Padé 逼近可以不同。

2025-07-21math.CAmath.CV

具有线性三角函数系数的Abel方程极限环的最大个数

Maximum number of limit cycles for Abel equation having coefficients with linear trigonometric functions

本文针对系数为线性三角函数的Abel方程，推广已有思想提出新准则，完全解决了Smale-Pugh问题的最简单情形，证明极限环最大个数为三。

2023-09-01math.CAmath.DS

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10-W-9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.

2003-12-16math-phmath.CAmath.MP

更多相关论文收起更多论文

Rational Limit Cycles of Abel Differential Equations

We study the number of rational limit cycles of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A(t)$ and $B(t)$ are real trigonometric polynomials. We show that this number is at most the degree of $A(t)$ plus one.

2023-02-21math.CA

Rational Solutions of Abel Differential Equations

We study the rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$ where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.

2021-09-16math.CA

The natural algorithmic approach of mixed trigonometric-polynomial problems

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function $\cos^2(x)$, and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.

2017-02-25cs.SCmath.CA

On the higher derivates of arctan

WWe give a rational closed form expression for the higher derivatives of the inverse tangent function and discuss its relation to Chebyshev polynomials, trigonometric expansions and Appell sequences of polynomials.

2017-06-16math.CA

Limits of elliptic hypergeometric integrals

In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.

2007-09-05math.CA

On the Abel-Nörlund-Voronoi summability and instability of rational maps

We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point on the Julia set of a given rational map.

2020-09-08math.CAmath.DS

Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials

Using an appropriate change of variable, the Schrödinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type $X_m$ exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric Scarf potentials and finite number of Hermitian and infinite number of non-Hermitian PT-symmetric hyperbolic Scarf potentials. The bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials. These infinite famili…

2013-03-12hep-thmath-phmath.CA

Large-degree asymptotics of rational Painleve-II functions. II

This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behavior. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve…

2014-06-02math-phmath.CAmath.MP