MA322 Proofs and Nmbers, MA323 Matrices and Probability or MA326 Matrices and Computing and MA553 Linear Algebra. Recommended but not essential MA324 Exploring Mathematics and MA554 Groups, Rings and Fields.
OverviewSystems of polynomial equations arise naturally in many applications of mathematics. This module focuses on methods for solving such systems and understanding the solutions sets. The key abstract concept is an ideal in a commutative ring and the fundamental computational concept is Buchberger's algorithm for computing a Groebner basis for an ideal in a polynomial ring. The syllabus includes: multivariate polynomials, Hilbert's Basis Theorem, monomial orders, division algorithms, Groebner bases, Hilbert's Nullstellensatz, elimination theory, linear equations over systems of polynomials, and syzygies.
This module appears in:
Up to 48 hours of lectures, examples classes and supervised problem-solving workshops.
Method of assessment
70% Examination, 30% Coursework
Cox, Little, OShea Ideals, Varieties and Algorithms, Springer, Undergraduate Texts in Mathematics, 1991
The intended subject specific learning outcomes
On successful completion of this module students will:
a) have acquired a broad understanding of modern methods of calculation with polynomials in several variables;
b) have increased their knowledge of the meaning and practice of solving systems of polynomial equations;
c) have learned how to formulate and prove statements about systems of polynomials in precise abstract algebraic language.
The intended generic learning outcomes
Students who successfully complete this module will have improved their ability to:
a) communicate their own ideas clearly and coherently in writing;
b) formulate and prove abstract mathematical statements, and appreciate their connections with concrete calculation.