沿极弧的横截多项式的双Lipschitz不变性
Bi-Lipschitz invariance of the transverse polynomial along polar arcs
- FPT University（富国大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究证明在双Lipschitz右等价下，约化全纯平面函数芽沿极弧的横截多项式在特定尺度范围内不变，并恢复增广牛顿多边形的双Lipschitz不变性，且横截多项式提供额外新信息。
AI中文摘要：
我们研究了约化全纯平面函数芽沿极弧的横截多项式。我们证明，在双Lipschitz右等价下，匹配的切向极弧具有相同的横截阶数，且其横截多项式在源与目标非零重标度下一致，对于每个有理尺度 $1<q<d(\gamma)$，其中 $d(\gamma)$ 是梯度峡谷度。该范围是精确的，因为结果在两个端点可能失效，即使在解析右等价下也是如此。作为应用，我们恢复了由Migus--Păunescu--Tibăr证明的增广牛顿多边形的双Lipschitz不变性。我们还给出一个例子，表明横截多项式包含极值不变量和增广牛顿多边形未检测到的新信息。
英文摘要：
We study transverse polynomials along polar arcs of reduced holomorphic plane function germs. We prove that, under bi-Lipschitz right equivalence, matched tangential polar arcs have the same transverse order and their transverse polynomials agree up to nonzero rescalings of the source and target, at every rational scale $1<q<d(γ)$, where $d(γ)$ is the gradient canyon degree. This range is sharp as the result can fail at both endpoints, even under analytic right equivalence. As an application, we recover the bi-Lipschitz invariance of the augmented Newton polygon proved by Migus--Păunescu--Tibăr. We also give an example showing that the transverse polynomial contains new information not detected by the polar-value invariants and augmented Newton polygons.
