Computational Algebraic Geometry

这项研究主要是出于对复杂性的考虑。 到 理解算法的复杂性，通常需要获得 交换代数的上下界。 过去几 近年来，在三个关键领域实现了新的跨越： Groebner基、Hilbert零点和Wu-Ritt特征线 集. 所采用的证明技术是新颖的，包含的想法， 具有更广泛的潜在应用。 这些和其他一些相关的 技术将进一步研究和新的算法， 这些想法将被设计出来。 另一个领域是有效算法的构建--对于GCD。 斯特姆序列，代数数计算。 Groebner基 和Wu-Ritt特征集 这里也有一些重要的进展 办妥了一批多年 进一步应用这些结果， 理想和簇的分解，以及更一般的 代数结构的类别将被追求。 一个没有得到太多关注的重要领域是 处理代数算法中数据结构的选择。 这似乎是一个非常令人兴奋的领域，因为计算对象 （多元多项式，理想）这里的结构要丰富得多 比任何传统的数据结构研究都要多。 将在这一领域开展研究，以解决 以下种类：各种数据结构的选择 计算代数算法，摊销和 散列技术到代数数据结构等。

The research is largely motivated by complexity considerations. To understand complexity of algorithms, it is often necessary to obtain good lower and upper bounds to commutative algebra. In the last few years, new bounds have been achieved in three key areas: namely, Groebner Bases, Hilbert Nullstellensatz and Wu-Ritt Characteristic Sets. The proof techniques employed are novel and contain ideas that have much wider potential applications. These and a few other related techniques will be further investigated and new algorithms based on these ideas will be devised. Another area is the construction of efficient algorithms -- for GCD. Sturm sequences, computations with algebraic numbers. Groebner bases and Wu-Ritt characteristic sets. Here also some important progress has been achieved. Further applications of these results to obtain decomposition of ideals and varieties, and extensions to more general classes of algebraic structures will be pursued. An important area that has not received much attention is the one dealing with the choice of data structures in algebraic algorithms . This seems a very exciting area because the computational objects (multivariate polynomials, ideals) here are much richer in structure than anything that has been traditionally studied in data structures. Research will be undertaken in this area to address issues of the following kind: Selection of data structures for various computational algebraic algorithms, Application of amortization and hashing techniques to algebraic data structures, etc.