CS 680 - Computational Algebra I

Prerequisite, MATH 211. A course in multivariate polynomials, their algebraic properties, and related algorithms for effective computations. After an introduction of the main concepts of the ring of single variable polynomials (polynomial ideals, unique factorization, division algorithm, similarities with the ring of integers), multivariable polynomials are defined. The course addresses the problem of defining order relations on the set of multivariate terms, and moves to the basic concepts of the theory of Gröbner bases. These include: the multivariate division algorithm as a generalization of the Gauss reduction algorithm for vector spaces; the Macaulay Basis theorem; viewing polynomials as rewrite rules; Buchberger’s algorithms for the construction of Gröbner bases for polynomial ideals; and the notion of syzygy. Throughout the course, students learn how to use a computer algebra software program to compute with polynomials and to implement the algorithms presented in class. (Offered as needed.) 3 credits