平面上的非完整理想与绝对因式分解
Non-holonomic Ideals in the Plane and Absolute Factoring
摘要：
我们研究了两个变量的左微分理想$J\subset F[\partial_x, \partial_y]$的{ \it非完整}超理想，其中$F$是特征为零的微分闭域。主要结果表明：由具有可分{ \it符号} $symb(P)$的算子$P$生成的主理想$J=< P>$，是二元齐次多项式，具有有限个极大非完整超理想.将这一陈述推广到具有可分符号的非完整理想$J$。作为应用，我们证明了在二阶算子$P$的情形下，理想$<P>$有无穷多个极大非完整超理想当且仅当$P$本质平凡.在三阶算子$P$的情况下，我们给出了$<P>$有有限个极大非完整超理想的几个充分条件。
英文摘要：
We study {\it non-holonomic} overideals of a left differential ideal $J\subset F[\partial_x, \partial_y]$ in two variables where $F$ is a differentially closed field of characteristic zero. The main result states that a principal ideal $J=< P>$ generated by an operator $P$ with a separable {\it symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a finite number of maximal non-holonomic overideals. This statement is extended to non-holonomic ideals $J$ with a separable symbol. As an application we show that in case of a second-order operator $P$ the ideal $<P>$ has an infinite number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In case of a third-order operator $P$ we give few sufficient conditions on $<P>$ to have a finite number of maximal non-holonomic overideals.
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