On successful completion of this module students will be able to
Compute the reduced Gröbner basis of an ideal in the polynomial 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.
Illustrate on examples the difference in applicability and scope between Gröbner bases and rewriting systems.
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, 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 variables 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.
Associative algebras. Free algebras. Algebras presented via generators and relations.
Gröbner bases and elimination in the commutative case: long division, Gauss-Jordan elimination, general case of solving systems of arbitrary polynomial equations. Finiteness, universal Gröbner bases.
Applications of Gröbner bases for commutative algebras outside abstract algebra.
Gröbner bases for associative algebras. Diamond Lemma. Examples.
Poincare-Birkhoff-Witt theorem via Gröbner bases.
Rewriting systems vs Gröbner bases.
Linear Algebra (MA1111 and MA1212), Fields, rings and modules (MA2215).
This module will be examined in a 2 hour examination in Trinity term.
The final mark is made up of 80% exam and 20% continuous assessment. For supplementals if required the supplemental exam will account for 100%.