Computational Algebraic Geometry

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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.