稀疏轮廓像的下陷、半正规化与导子
Descent, Seminormalization, and Conductors of Sparse Profile Images
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中文总结 AI 辅助
本文为稀疏轮廓像建立下陷理论，通过整纤维完备化与半群模型分类齐次及非周期轮廓，计算导子与深度，并证明正规化等不决定实际像。
中文摘要 AI 辅助
我们为稀疏分解轮廓像发展了一套下陷理论，探讨正规化上的哪些函数下陷到实际像。整纤维完备化对齐次掩模给出傅里叶零分类，并具有锐利的稳定逆命题；对混合非周期轮廓，在所标稳定范围内给出傅里叶线分类。对齐次轮廓，有界整数关系决定几何纤维与半正规化，而显式半群模型在大量族中计算实际像代数、导子与深度。对单点补轮廓，精确的一阶条件决定有限输出代数、导子以及通过第一非正规边界的射影下陷；在光滑边界范围和第一非正规层的全零点处计算局部类型。我们还证明了饱和单位根特化定理，并给出分离例子，表明正规化、自然极化、点纤维、半正规化乃至源导子都不必决定实际像。
英文摘要
We develop a descent theory for sparse factorization-profile images, asking which functions on a normalization descend to the actual image. Whole-fiber completions yield a Fourier-zero classification for homogeneous masks, with a sharp stable converse, and a Fourier-line classification for mixed aperiodic profiles in the stated marked stable ranges. For homogeneous profiles, bounded integer relations determine geometric fibers and seminormalization, while explicit semigroup models compute actual image algebras, conductors, and depth in substantial families. For singleton--complement profiles, exact one-jet conditions determine the finite-output algebra, conductor, and projective descent through the first nonnormal boundary; the local type is computed in the smooth-boundary range and at the total-zero point of the first nonnormal layer. We also prove a saturated root-of-unity specialization theorem and give separating examples showing that normalization, natural polarizations, point fibers, seminormalization, and even the source conductor need not determine the actual image.
发表机构
- School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
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