Studies on Cores of Ideals and Blowup Algebras

本课题主要研究交换代数及其与计算代数和代数几何的相互作用。该提案的重点是爆破代数理论。这些代数自然而然地出现在代数和几何的许多结构中。例如，它们出现在奇点分解的过程中。他们研究的一个重要工具是理想的约化概念。在这种情况下出现了几个键不变量，例如约化数，它控制着blowup代数的Cohen-Macaulay性质。为了同时研究所有约化，我们考虑理想的核心，定义为这些约化的交集。这个对象与系数理想、伴随理想和乘子理想有关，在Brian\c{c}on-Skoda型定理中起着至关重要的作用。这一建议的一个主要目的是给出单项式理想的核的组合描述，阐明核与约化数、第一系数理想和伴随或乘子理想的关系，研究具有任意特征的核，并建立以前仅在特征零时才知道的公式。此外，该项目还研究了理想的积分闭包等其他领域。环和理想的积分扩张和积分闭包的概念是交换代数的核心。该项目的目标之一是找到计算积分闭包复杂性的先验度量。现实生活中的问题通常涉及许多未知参数，这些参数与无法精确求解的方程相关。然而，使用交换代数方法，可以获得关于潜在解集的许多有价值的描述性信息，即使不是确切的解本身。在过去的二十年里，随着重要猜想的解决，交换代数取得了巨大的活跃和成功，它应用于计算机科学、密码学、编码理论、机器人学、模式识别和理论物理等不同领域，并发现了与数学其他部分的意想不到的联系，从拓扑学到组合学，从计算机代数到统计学。

This project is concerned with problems from commutativealgebra and its interactions with computational algebra andalgebraic geometry. The focus of the proposal is on the theory ofblowup algebras. These algebras appear naturally in manyconstructions both in algebra and geometry. For example they arisein the process of resolution of singularities. An essential tool intheir study is the concept of reduction of an ideal. Several keyinvariants emerge in this context such as the reduction number,which among other things controls the Cohen-Macaulay property ofblowup algebras. To study all reductions at once one considers thecore of an ideal, defined as the intersection of these reductions.This object, related to coefficient, adjoint and multiplier ideals,plays a crucial role in Brian\c{c}on-Skoda type theorems. A mainthrust of this proposal is to give a combinatorial description ofthe core of monomials ideals, to clarify the connection of the corewith the reduction number, the first coefficient ideal and theadjoint or multiplier ideal, to investigate the core in arbitrarycharacteristic and to establish formulas that were previously knownonly in characteristic zero. In addition, the project also studiesother areas such as integral closures of ideals. The concepts ofintegral extensions and integral closures of rings and ideals arecentral to much of commutative algebra. One of the goals of theproject is to find a priori measures for the complexity of computingintegral closures.Often real life problems involve many unknown parameters that arerelated by equations which are impossible to solve exactly.Nevertheless, using commutative algebra methods, much valuabledescriptive information can be gained about the potential sets ofsolutions, if not the exact solutions themselves. Commutativealgebra has seen a great deal of activity and success over the pasttwo decades with the solution of important conjectures, itsapplication to diverse fields such as computer science,cryptography, coding theory, robotics, pattern recognition andtheoretical physics, and the discovery of unexpected connections toother parts of mathematics, ranging from topology to combinatoricsand from computer algebra to statistics.