The proposed research belongs to the interface between mathematics and computer science. Professor Gao will investigate efficient algorithms for polynomial systems. Topics include large systems of linear equations, polynomial factorization, primary decomposition, and Hilbert irreducibility theorems. These topics are closely interrelated and are fundamentally important in algebra, number theory, and symbolic computation.
Algebra and number theory are the oldest branches of mathematics, and polynomial systems play an important role throughout the history of mathematics. With the increasing use of computers in science and engineering, efficient solution of various problems related to polynomials are of vital importance. In addition, many cryptographic schemes for protecting sensitive digital information (from cellular phone communications, bank transactions, e-commerce to national security) are based on algebraic structures, and their security relies on various hard problems in algebra and number theory. The current research studies a sample of problems related to these applicati
