对称矩阵的幺模族
Unimodular families of symmetric matrices
摘要：
本文引入了对称矩阵值映射芽之间的保体积等价，它是Bruce的$\mathcal{G}$ -等价的幺模形式.从与$\mathcal{G}$ -等价有关的分类中导出幺模分类的关键概念是对称拟齐性，它是Bruce，Goryunov和Zakalyukin在推论~2.1（ii）中关于$2 \times 2$对称矩阵-值映射芽的条件的推广.如果一个$\mathcal{G}$ -等价类包含一个对称的拟齐次表示，则该类与相对于保体积等价的类重合（直到在基域为实数的情况下的方向反转同构）。通过使用，我们表明，所有的简单类相对于$\mathcal{G}$ -等价在布鲁斯的列表中，与那些相对于体积保持等价。然后，我们分类映射芽从平面的集合$2 \times 2$和$3 \times 3$实对称矩阵的corank最多$1$和$\mathcal{G}_e$ -余维小于$9$，我们显示了一些标准形式分裂成两个不同的单模奇点。本文给出几个例子说明非单性并不意味着非对称拟齐性，而且映射芽对称拟齐性的条件比映射芽的每个分支拟齐性的条件强.给出了一个相对于$\mathcal{G}$的非对称拟齐次规范形及其相应的形式么模规范形的例子。
英文摘要：
We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence. The key concept to deduce unimodular classification out of classification relative to $\mathcal{G}$-equivalence is symmetrical quasi-homogeneity, which is a generalization of the condition for a $2 \times 2$ symmetric matrix-valued map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a $\mathcal{G}$-equivalence class contains a symmetrically quasi-homogeneous representative, the class coincides with that relative to the volume-preserving equivalence (up to orientation reversing diffeomorphism in case if the ground field is real). By using that we show that all the simple classes relative to $\mathcal{G}$-equivalence in Bruce's list coincides with those relative to the volume preserving equivalence. Then, we classify map-germs from the plane to the set of $2 \times 2$ and $3 \times 3$ real symmetric matrices of corank at most $1$ and of $\mathcal{G}_e$-codimension less than $9$ and we show some of the normal forms split into two different unimodular singularities. We provide several examples to illustrate that non simplicity does not imply non symmetrical quasi-homogeneity and the condition that a map-germ is symmetrically quasi-homogeneous is stronger than one that each component of the map-germ is quasi-homogeneous. We also present an example of non symmetrically quasi-homogeneous normal form relative to $\mathcal{G}$ and its corresponding formal unimodular normal form.
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