关于具有六边形网解的隐式常微分方程
On implicit ODEs with hexagonal web of solutions
摘要：
一个隐式常微分方程的解形成一个网。已经为三次常微分方程的3网的解决方案有一个非平凡的局部不变，即曲率形式。因此，如果对常微分方程的类没有任何限制，隐式常微分方程的任何局部分类必然具有泛函模。这里讨论的最对称的情况下，六边形3-网的解决方案，即曲率假设为零相同。在一些自然正则性假设下，建立了一个有限规范形表。这些假设的几何意义是，在1-喷流空间中由常微分方程定义的曲面是光滑的，并且犯罪者是该曲面投影到平面上的临界集。
英文摘要：
Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometrical meaning of these assumptions is that the surface, defined by ODE in the space of 1-jets, is smooth as well as the criminant, which is the critical set of this surface projection to the plane.
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