Details of Grant 

EPSRC Reference:	GR/T04458/01
Title:	Diophantine equations in roots of unity
Principal Investigator:	Smyth, Professor C
Other Investigators:	Macintyre, Professor A
Researcher Co-Investigators:
Project Partners:
Department:	Sch of Mathematics
Organisation:	University of Edinburgh
Scheme:	Standard Research (Pre-FEC)
Starts:	18 January 2005	Ends:	17 January 2008	Value (£):	154,359
EPSRC Research Topic Classifications:	Algebra & Geometry Logic & Combinatorics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
A central theme in mathematics is finding solutions of equations. In algebraic geometry, the equations to be solved are systems of polynomial equations in several variables, and the variables are assumed to lie in some field of numbers. For classical Diophantine equations, thevariables are restricted to be integers. Both these topics have a long history, and are very active areas of current research. The topicof the proposal concerns the solution of equations in roots of unity, where some power of each of the variables is 1. This subject has been comparatively neglected, but turns out to have a surprising range of applications. We plan to build on some earlier work in this area, which concerned the case of a curve (one equation in two variables) . We shall start by studying some other special cases such as surfaces (one equation in three variables) and space curves (two equations in three variables), before working on more general systems of equations. It isknown from the work of W. Schmidt and others that all solutions can be described in terms of a finite number of parametric families called torsion cosets. The central aim of our project is the study of these cosets. We plan to design an efficient algorithm for finding them, to get goodupper bounds for the maximum number there can be, and also find good bounds for the so-called exponent of the coset, which describes the rootst of unity involved in its definition. The methods we plan to use are both number-theoretic and model-theoretic. The success of Hrushovski in applying model-theoretic methods to the Manin-Munford conjecture in diophantine geometry encourages us to expect that they will also prove useful and fruitful for our topic. `
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