单项式理想的相伴素、见证元与omega不变量
Associated primes, witnesses, and omega invariants of monomial ideals
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中文总结 AI 辅助
本文引入诺特环中理想的omega不变量（相伴素理想个数），通过单项式见证元与矩阵方法刻画单项式理想及其幂的相伴素，给出无需准素分解的公式与界，并推广至边理想。
中文摘要 AI 辅助
我们引入并研究了诺特交换环中真理想的omega不变量，该不变量定义为理想的相伴素理想的数量。我们的主要目标是研究单项式理想及其幂的该不变量。我们通过单项式见证元刻画相伴素理想，并给出一种从极小生成元的指数向量构造此类见证元的算法程序。这些结果导出了omega不变量的显式公式和界，而无需计算准素分解。我们还利用不可约分解和Alexander对偶建立了替代描述。开发了一种基于矩阵的方法，直接从原理想的指数矩阵检测单项式理想幂的相伴素理想。我们还研究了见证元在从$I^n$到$I^{n+1}$过程中的行为，并推导了图的边理想的相应结果。
英文摘要
We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from $I^n$ to $I^{n+1}$ and derive corresponding results for edge ideals of graphs.
