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The method of Puiseux series and invariant algebraic curves
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2021-02-18 , DOI: 10.1142/s0219199721500073
Maria V. Demina 1
Affiliation  

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems xt = y, yt = f(x)y g(x) with deg f <deg g < 2deg f + 1. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with deg f <deg g < 2deg f + 1 can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

中文翻译:

Puiseux级数和不变代数曲线的方法

导出了平面内多项式动力系统不可约不变代数曲线相关的辅因子的显式表达式。得到了平面内多项式动力系统具有有限条不可约不变代数曲线的充分条件。所有这些结果都应用于 Liénard 动力系统X = 是的,是的 = -F(X)是的 - G(X) F < G < 2 F + 1. 找到了它们不可约不变代数曲线和辅因子的一般结构。证明了 Liénard 动力系统具有 F < G < 2 F + 1最多可以同时有两条不同的不可约不变代数曲线,因此不能与有理一阶积分积分。
更新日期:2021-02-18
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