Decompositions of Ideals

理想的分解

• 批准号: 0200420
• 负责人: Irena Swanson
• 金额: $ 9.73万
• 依托单位: New Mexico State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200420&HistoricalAwards=false
• 关键词: Decompositions; Ideals

The PI will continue her investigations on the decompositions of ideals and how different operations on the ideals interact with the decompositions. The particular operations in the proposal are integral closure, tight closure, adjoints, Rees valuations, symbolic powers, Groebner bases. The PI works in the area of mathematics known as commutative algebra, with applications to the area of algebraic geometry. In particular, there are algebraic constructs that describe geometric objects, such as spheres, cones, cylinders, etc. The (non) "smoothness" as well as many other geometric properties of a geometric object are reflected in the properties of its corresponding algebraic construct, which is useful because the latter is much more susceptible to manipulation. A familiar example might be the prime factorization of integers or polynomials, such as used in cryptography, but in more general situations the corresponding "factorizations" are more complex and highly non-unique.

PI 将继续研究理想的分解以及理想的不同操作如何与分解相互作用。 该提案中的具体操作是积分闭合、紧闭合、伴随、Rees 估值、符号幂、Groebner 基。 PI 的工作领域为交换代数数学领域，并应用于代数几何领域。 特别是，存在描述几何对象的代数构造，例如球体、圆锥体、圆柱体等。几何对象的（非）“平滑性”以及许多其他几何属性反映在其相应代数构造的属性中，这是有用的，因为后者更容易受到操纵。 一个熟悉的例子可能是整数或多项式的素因式分解，例如在密码学中使用的，但在更一般的情况下，相应的“因式分解”更复杂且高度非唯一。