MITACS Seminar Series on Mathematics of Computer Algebra and Analysis
Regular Chains: Theory and Practice
Paul Vrbik, Department of Computer Science, University of Western Ontario
Abstract: Regular chains are one of the only real alternatives to Groebner bases (GB) for solving systems of polynomials. Using regular chains instead of GB is becoming more common; evidenced by regular chains supplanting Groebner bases in Maple's "solve" command for difficult input polynomial systems. Why? I start with the basic notions of primary decomposition, saturation ideals, and triangular sets. Then refine these ideas to discuss the fundamental theoretical properties of regular chains and how one uses them to decompose polynomial systems. Once a sufficient theoretical groundwork is laid I will demonstrate how the recursive definitions lend themselves naturally to algorithms and show that these algorithms produce output preferable to that of a GB.