标题： 不定度规的Toda格 lattice II：同谱流形的拓扑结构 显示英文标题

标题： Toda lattices with indefinite metric II: Topology of the iso-spectral manifolds

摘要： 我们研究了具有实特征值的三对角Hessenberg矩阵的等谱实流形。 这些流形由作者引入的不定Toda格链方程的等谱流来描述[Physica, 91D (1996), 321-339]。 这些Toda格链由具有哈密顿量 $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1} s_ks_{k+1} \exp(x_k-x_{k+1})$ 的 $2^{N-1}$ 个不同系统组成，其中 $s_i=\pm 1$。 通过添加无穷大来紧化这些流形，除了所有 $s_is_{i+1}=1$ 都为特定情况之外，这些Toda流会在有限时间内爆炸。 结果表明，当 $N>2$ 时，这些流形是非定向的，并且对称群是 $(\ZZ_2)^{N-1}$ 和置换群 $S_N$ 的半直积。 这些性质与Davis和Januszkiewicz引入的“小覆盖”相一致[Duke Mathematical Journal, 62 (1991), 417-451]。 作为我们构造的一个推论，我们给出了由Hankel行列式生成的指数多项式系统的零点总数公式。 显示英文摘要

摘要： We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different systems with hamiltonians $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1} s_ks_{k+1} \exp(x_k-x_{k+1})$, where $s_i=\pm 1$. We compactify the manifolds by adding infinities according to the Toda flows which blow up in finite time except the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to be nonorientable for $N>2$, and the symmetric group is the semi-direct product of $(\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction, we give a formula on the total numbers of zeroes for a system of exponential polynomials generated as Hankel determinant.