论文标题
简单的派生及其图像
Simple derivations and their images
论文作者
论文摘要
在本文中，我们证明了派生$ d = y \ partial_x+（a_2（x）y^2+a_1（x）y+a_0（x））\ partial_y $ of $ k [x，y] $ a_2（x），a_1（x），a_1（x），a_0（x），a_0（x），a_0（x） $ a_0（x）\ in K^*$，$（2）$ $ $ $ $ $ a_1（x）\ geq1 $或$°a_2（x）\ geq1 $，$（3）$，存在$ l \ in k^*$中的$ l \ in k^*$，因此$ l \ y_2（x）= la_1（x）= la_1（x）-L^2a_0（x）-l^2a_0（x）$。此外，我们证明了派生的图像$ d = \ partial_x+{\ sum_ {i = 1}^nγ_i（x）y_i^{k_i}} {\ partial_i} $是Mathieu-Zhao Space Iff $ d $是本地有限的。此外，我们证明了派生的图像$ d = {\ sum_ {i = 1}^nγ_iy_i^y_i^{k_i}} {\ partial_i} $ of $ k [y_1，\ ldots，y_n] $ n \ geq 2 $。
In the paper, we prove that the derivation $D=y\partial_x+(a_2(x)y^2+a_1(x)y+a_0(x))\partial_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^*$, $(2)$ $°a_1(x)\geq1$ or $°a_2(x)\geq1$, $(3)$ there exist no $l\in K^*$ such that $a_2(x)=la_1(x)-l^2a_0(x)$. In addition, we prove that the image of the derivation $D=\partial_x+{\sum_{i=1}^n γ_i(x) y_i^{k_i}}{\partial_i}$ is a Mathieu-Zhao space iff $D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum_{i=1}^n γ_i y_i^{k_i}}{\partial_i}$ of $K[y_1,\ldots,y_n]$ is a Mathieu-Zhao space iff $k_i\leq 1$ for all $1\leq i\leq n$, $n\geq 2$.
