Studies on Cores of Ideals and Blowup Algebras

这个项目关注的是交换代数的问题及其与计算代数和代数几何的相互作用。该提案的重点是理论ofblowup代数。这些代数自然地出现在代数和几何的许多构造中。例如，它们出现在奇点的解决过程中。在他们的研究中一个重要的工具是一个理想的还原的概念。几个关键不变量出现在这种情况下，如减少数，其中除其他事项外，控制科恩-麦考利性质的爆破代数。为了同时研究所有的约化，我们考虑一个理想的核，它被定义为这些约化的交集，这个对象与系数、伴随和乘子理想有关，在BrianOn-Skoda型定理中起着至关重要的作用。这个建议的主旨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理想或乘子理想的联系，研究任意特征的核，建立以前只知道特征零的公式。此外，该项目还研究了理想的积分闭包等其它方面的问题。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的复杂性的先验度量。通常，真实的生活问题涉及许多未知参数，这些参数由不可能精确求解的方程相关联。然而，使用交换代数方法，即使不是精确解本身，也可以获得关于潜在解集的许多有价值的描述信息。在过去的二十年里，交换代数在解决重要问题方面取得了巨大的成功，它在计算机科学、密码学、编码理论、机器人技术、模式识别和理论物理等不同领域的应用，以及与数学其他部分的意外联系的发现，从拓扑学到组合数学，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.