Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory.
Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory.
复制标题
嘉当移动框架方法的归纳方法及其在经典不变理论中的应用。
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
I. Kogan
中科院分区：
文献类型：
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作者：
I. Kogan
本论文致力于研究李群作用下子流形的等价性问题的嘉当移动标架方法的算法实现。我们采用了一个一般的定义，一个移动框架作为一个等变的映射，从空间的子流形的组本身，并介绍了两个算法，这简化了这种映射的建设。第一个算法是适用的，当组因子作为两个子群$G=BA$的产品，允许我们使用移动框架和微分不变量的组$A$和$B$，以构建一个移动框架和微分不变量的$G$。这种方法产生的$G$和它的子群的不变量之间的关系。我们用平面上的射影变换、仿射变换和欧几里德变换的群来说明算法。我们还介绍了一个递归算法，允许，提供了组行动满足一定的条件，构造微分不变量的顺序顺序，在每一步规范化越来越多的组参数，在年底获得一个移动的框架，为整个集团。该算法的发展是由于将移动标架法应用于变量线性变化下多项式的等价性和对称性问题而产生的。在复或真实的情况下，这些问题可以简化，并在理论上完全解决的问题的等价子流形。然而，它的解决方案涉及基于Grobner基础计算的算法，由于其复杂性，并不总是可行的。尽管如此，还是得到了一些有趣的新结果，如三元三次函数及其对称群的分类，三元齐次多项式等价于x^n+y^n+z^n的充要条件等。
This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algorithms, which simplify the construction of such maps. The first algorithm is applicable when the group factors as a product of two subgroups $G=BA$, allowing us to use moving frames and differential invariants for the groups $A$ and $B$ in order to construct a moving frame and differential invariants for $G$. This approach produces the relations among the invariants of $G$ and its subgroups. We use the groups of the projective, the affine and the Euclidean transformations on the plane to illustrate the algorithm. We also introduce a recursive algorithm, allowing, provided the group action satisfies certain conditions, to construct differential invariants order by order, at each step normalizing more and more of the group parameters, at the end obtaining a moving frame for the entire group. The development of this algorithm has been motivated by the applications of the moving frame method to the problems of the equivalence and symmetry of polynomials under linear changes of variables. In the complex or real case these problems can be reduced and, in theory, completely solved as the problem of the equivalence of submanifolds. Its solution however involves algorithms based on the Grobner basis computations, which due to their complexity, are not always feasible. Nevertheless, some interesting new results were obtained, such as a classification of ternary cubics and their groups of symmetries, and the necessary and sufficient conditions for a homogeneous polynomial in three variables to be equivalent to $x^n+y^n+z^n.$