多项式与正规矩阵的拟解析多参数扰动
Quasianalytic multiparameter perturbation of polynomials and normal matrices
摘要：
研究了系数属于$C^\infty$ -函数的某个子环$\mathcal C$的固定次数复一元一元多项式$P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$的多参数族根的正则性.我们要求$\mathcal C$包括多项式但不包括平坦函数（拟解析性），并且在复合、求导、除以坐标和取逆下是封闭的。例子是拟解析Denjoy-Carleman类，特别是一类真正的解析函数$C^ω$。本文证明了参数空间存在一个局部有限覆盖$\{π_k\}$，其中每个$π_k$是一个由许多个$\mathcal C$ -映射组成的复合，每个$\mathcal C$ -映射要么是一个具有光滑中心的局部爆破，要么是一个局部幂代换（在$x \mapsto (\pm x_1^{γ_1},...,\pm x_q^{γ_q})$，$γ_i \in \mathbb N_{>0}$给出的坐标中），使得对于每个$k$，多项式族$P øπ_k$允许其根的$\mathcal C$ -参数化.如果$P$是双曲的（所有根都是实的），那么局部爆破就足够了。利用这个去奇异化结果，我们证明了$P$的根可以被$SBV_{loc}$ -函数参数化，这些函数的经典梯度几乎处处存在，并且属于$L^1_{loc}$ .通常，对于任何$1 < p \le \infty$，根在$L^p_{loc}$中不能具有梯度。根也不能位于$W_{loc}^{1,1}$或$VMO$中。我们得到了正规矩阵的$\mathcal C$ -族的特征值和特征向量的相同的正则性。进一步的推论是每个连续的次解析函数都属于$SBV_{loc}$。
英文摘要：
We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$ of $C^\infty$-functions. We require that $\mathcal C$ includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy--Carleman classes, in particular, the class of real analytic functions $C^ω$. We show that there exists a locally finite covering $\{π_k\}$ of the parameter space, where each $π_k$ is a composite of finitely many $\mathcal C$-mappings each of which is either a local blow-up with smooth center or a local power substitution (in coordinates given by $x \mapsto (\pm x_1^{γ_1},...,\pm x_q^{γ_q})$, $γ_i \in \mathbb N_{>0}$), such that, for each $k$, the family of polynomials $P øπ_k$ admits a $\mathcal C$-parameterization of its roots. If $P$ is hyperbolic (all roots real), then local blow-ups suffice. Using this desingularization result, we prove that the roots of $P$ can be parameterized by $SBV_{loc}$-functions whose classical gradients exist almost everywhere and belong to $L^1_{loc}$. In general the roots cannot have gradients in $L^p_{loc}$ for any $1 < p \le \infty$. Neither can the roots be in $W_{loc}^{1,1}$ or $VMO$. We obtain the same regularity properties for the eigenvalues and the eigenvectors of $\mathcal C$-families of normal matrices. A further consequence is that every continuous subanalytic function belongs to $SBV_{loc}$.
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