发表机构
Tulane University（杜兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Banach空间上实解析泛函的最优Lojasiewicz-Simon指数是其Lyapunov-Schmidt奇点的内蕴不变量，并给出其有理性与显式计算公式，应用于矩阵、Yang-Mills及共振Dirichlet能量。
AI 中文摘要
我们证明，在自然的Fredholm和配对相容性假设下，Banach空间上实解析泛函的最优Lojasiewicz-Simon指数是其有限维Lyapunov-Schmidt奇点的内蕴不变量。更精确地说，Lyapunov-Schmidt约化保持整个容许指数集，而Hessian核上的约化芽在切于恒等的解析右等价下是内蕴的。然后我们将该不变量与由约化能量的Jacobian理想及其主能量理想组成的对子的实相对Lojasiewicz指数等同。因此，最优指数可通过解析弧和可除比率的有限最大值等价描述；特别地，它是有理数。在Newton非退化情形下，这些公式给出显式的单项式表达式。我们将该理论应用于几类奇异能量。对于实矩阵的所有$k\times k$子式的平方和$Q_k$，我们确定其关联理想对子直至实积分闭包，并在秩$s<k$的矩阵处获得精确的局部指数$$1-\frac1{2(k-s)}$$。对于Yang-Mills能量，我们证明对每个非交换紧结构群，$\mathbb T^2$上的乘积平坦联络具有最优指数$3/4$。对于$\mathrm{SU}(2)$，行列式约化将该值推广到每个$\mathbb T^d$上的乘积平坦联络，而足够小的非零常数平坦联络具有指数$1/2$。最后，对于共振半线性Dirichlet能量，我们识别高阶共振障碍，为简单共振建立全阶障碍阶梯，并在$(0,\pi)$上获得完整的奇偶依赖分类。
英文摘要
We prove that the optimal Lojasiewicz--Simon exponent of a real-analytic functional on a Banach space is an intrinsic invariant of its finite-dimensional Lyapunov--Schmidt singularity under natural Fredholm and pairing-compatible hypotheses. More precisely, Lyapunov--Schmidt reduction preserves the entire set of admissible exponents, while the reduced germ on the kernel of the Hessian is intrinsic up to analytic right-equivalence tangent to the identity. We then identify this invariant with a real relative Lojasiewicz exponent of the pair consisting of the Jacobian ideal of the reduced energy and its principal energy ideal. Consequently, the optimal exponent admits equivalent descriptions by analytic arcs and by a finite maximum of divisorial ratios; in particular, it is rational. In Newton-nondegenerate cases, these formulas yield explicit monomial expressions. We apply the theory to several classes of singular energies. For the sum $Q_k$ of the squares of all $k\times k$ minors of a real matrix, we determine the associated ideal pair up to real integral closure and obtain the exact local exponent $$1-\frac1{2(k-s)}$$ at matrices of rank $s<k$. For Yang--Mills energy, we prove that the product flat connection on $\mathbb T^2$ has optimal exponent $3/4$ for every nonabelian compact structure group. For $\mathrm{SU}(2)$, the determinantal reduction extends this value to product flat connections on every $\mathbb T^d$, while sufficiently small nonzero constant flat connections have exponent $1/2$. Finally, for resonant semilinear Dirichlet energies, we identify higher-order resonant obstructions, establish an all-order obstruction ladder for simple resonance, and obtain a complete parity-dependent classification on $(0,π)$.
Comments32 pages, no figures
