Polynomials in Several Variables - MAST7027

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Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2022 to 2023
Autumn Term 7 15 (7.5) James Shank checkmark-circle


This module provides a rigorous foundation for the solution of systems of polynomial equations in many variables. In the 1890s, David Hilbert proved four ground-breaking theorems that prepared the way for Emmy Nöther's famous foundational work in the 1920s on ring theory and ideals in abstract algebra. This module will echo that historical progress, developing Hilbert's theorems and the essential canon of ring theory in the context of polynomial rings. It will take a modern perspective on the subject, using the Gröbner bases developed in the 1960s together with ideas of computer algebra pioneered in the 1980s.

Indicative syllabus:
• Multivariate polynomials, monomial orders, division algorithm, Gröbner bases;
• Hilbert's Nullstellensatz and its meaning and consequences for solving polynomials in several variables;
• Elimination theory and applications;
• Linear equations over systems of polynomials, syzygies.

Level 7 students will cover additional topics such as polynomial maps between varieties.


Contact hours

Total contact hours: 42
Private study hours: 108
Total study hours: 150

Method of assessment

80% examination, 20% coursework

Indicative reading

Adams, Loustaunau, An introduction to Gröbner bases, AMS, 1994
Cox, Little, O'Shea, Ideals, Varieties and Algorithms, Springer, Undergraduate Texts in Mathematics, 1991
Hibi, Gröbner bases: Statistics and Software Systems, Springer, 2013

Learning outcomes

The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of polynomials in several variables;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: solution sets
for systems of polynomial equations and the corresponding ideals in the ring of polynomials;
3 apply a range of concepts and principles of polynomials in several variables in loosely defined contexts, showing good judgment in the selection and application of tools
and techniques;
4 make effective and well-considered use of computer calculation of Gröbner bases.

The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.


  1. Credit level 7. Undergraduate or postgraduate masters level module.
  2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
  3. The named convenor is the convenor for the current academic session.
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