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Level: BSc, MSci, MSc
Title: Newton polytopes and factorisation
Supervisor:

Dr Alex Fink

Research Area: Algebra
Description:

A polytope is the generalisation of a polygon or polyhedron to any dimension. Given a polynomial in multiple variables, the Newton polytope is a polytope obtained from the set of exponent vectors of its terms. Newton polytopes transform in a recognisable way when two polynomials are multiplied, so inspecting them gives an algorithm that can detect irreducibility of polynomials. The version of this algorithm for polynomials in one variable over a field with a nonarchimedean valuation is the case Newton did (though not in this language!) and the reason for honouring him with the name. The aim of this project is to give an exposition, and ideally an implementation, of this algorithm.

For BSc, focussing on the two-dimensional case would be sufficient.

Further Reading:
  • S. Gao, A. G. B. Lauder, Decomposition of polytopes and polynomials, Discrete Comput. Geom. 26 (2001) 89–104.
Plus a book on polyhedral geometry, such as:
  • B. Grünbaum, Convex Polytopes. Second edition. Graduate Texts in Mathematics, Vol. 221, Springer-Verlag, New York, 2003.
  • G. M. Ziegler, Lectures on Polytopes. Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, New York, 1995.
Key Modules:

MTH4104 Introduction to Algebra

MTH5112 Linear Algebra I or MTH6140 Linear Algebra II
Other Information:
Current Availability: Yes