On a classification of polynomial differential operators with respect to the type of first integrals

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• 作者：Jinzhi Lei ^{jzlei@tsinghua.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34A05 ; 34A34 ; 12H05
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 February 2016
• 年：2016
• 卷：260
• 期：3
• 页码：1993-2025
• 全文大小：418 K

文摘

This paper gives a classification of polynomial differential operators X=X₁(x₁,x₂)δ₁+X₂(x₁,x₂)δ₂$\mathcal{X}=X_1(x_1,x_2){\delta }_1+X_2(x_1,x_2){\delta }_2$(δ_i=∂/∂x_i)$({\delta }_i=\partial /\partial x_i)$. The classification is defined through an order derived from X$\mathcal{X}$. Let X=Xy$X=\mathcal{X}y$ be the associated differential polynomial, the order is defined as the order of a differential ideal Λ that is an essential extension of {X}$\{X\}$. The main result shows the order can only be four possible values: 0, 1, 2, 3, or ∞. Furthermore, when the order is finite, the essential extension Λ={X,A}$\mathrm{\Lambda }=\{X,A\}$, where A   is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of X$\mathcal{X}$. When the order is infinite, the extension Λ is identical with {X}$\{X\}$. In addition, if, and only if, the order is 0, 1, or 2, the associated polynomial differential equation has Liouvillian first integrals. Examples and connections with Godbillon–Vey sequences are also discussed.