Computational Algebraic Geometry

这项研究在很大程度上是出于复杂性考虑。为了理解算法的复杂性，通常需要得到好的交换代数的上下界。在过去的几年里，在三个关键领域取得了新的界：即Groebner基，Hilbert Nullstellensatz和Wu-Ritt特征集。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技术将被进一步研究，并将基于这些想法设计新的算法。另一个领域是构建高效的算法--用于GCD。Sturm序列，用代数数进行计算。Groebner基和Wu-Ritt特征集。在这方面也取得了一些重要进展。进一步应用这些结果来获得理想和簇的分解，以及扩展到更一般的代数结构类。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选择的领域。这似乎是一个非常令人兴奋的领域，因为这里的计算对象(多元多项式、理想)在结构上比传统上在数据结构中研究的任何对象都丰富得多。将在这一领域进行研究，以解决以下问题：为各种计算代数算法选择数据结构，将摊销和散列技术应用于代数数据结构等。

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.