On successful completion of this module students will be able to:
Compute the reduced Gröbner basis of an ideal in the polynominal algebra.
Prove the existence and uniqueness of the reduced Gröbner basis, and justify the validity of Buchberger's algorithm.
Use noncommutative Gröbner bases to determine the growth rate for an associative algebra, and compute Hilbert series for an algebra with monomial relations.
Use available computer software to compute Gröbner bases.
State and prove some termination criteria for computing Gröbner bases for noncommutative algebras.
This module is intended as an introduction to an important computational method of algebra, Gröbner bases (for 'systems of equations' in a reasonably wide sense). This method, implicit in works of various mathematicians for a long time,(since at least 1900), has only been made into a general theory as recently as in 1965. It can be viewed as a generalisation of both long division and Gaussian elimination (of unknowns) from linear equations, leading to efficient methods of solving systems of polynomial equations, and hence applicable in a range of subjects across both pure mathematics and ''real life'' applications like robotics and image processing. The module will set out theoretical foundations for this theory, provide the students with examples to explore, and outline some applications of Gröbner bases in natural sciences.
Gröbner bases and elimination in the commutative case: long division, Gauss-Jordan alimination, Buchberger's algorithm. Speeding up the algorithm: ''Triangle lemma''.
Using Gröbner bases for solving systems of arbitrary polynomial equations. ''Shape lemma.''
Dickson's lemma. Finiteness, universal Gröbner bases. Classification of monomial orders.
Gröbner bases for associative algebras. Diamond lemma. Examples. Exotic monomial orders. Criteria of termination in the non-commutative case.
Applications of commutative and noncommutative Gröbner bases.
This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. Supplemental exams if required will consist of 100% exam.