The main aim of our project was to study solutions of polynomial equations in
roots of unity. It was conjectured by Serge Lang and proved by Michel Laurent
that all such solutions can be described in terms of a finite number of
parametric families called maximal torsion cosets. We obtain new explicit
upper bounds for the number of maximal torsion cosets on an algebraic
subvariety of the complex multiplicative algebraic n-torus G_m^n.
In contrast to earlier
work that gives the bounds of polynomial growth in the maximum total
degree of defining polynomials, the proofs of our results are constructive.
This allowed us to obtain a new algorithm for determining maximal torsion
cosets on an algebraic subvariety of G_m^n.
Multiplying a number n by 1 doesn't change it. A root of unity is a number w such that you can leave n unchanged by multiplying n by w some whole number of times. For instance, a tenth root of unity is a (complex) number w such that when you multiply n by w ten times, n is unchanged.
Diophantine equations are general families of equations in variables x,y,z,... with integer coefficients. We are interested in when the equations are true (both sides equal) when the variables take on values that are roots of unity. Typically, these roots of unity collect in infinite families called maximal torsion cosets, and there are only a finite number of such families. But how many? The main aim of the project was to find how many maximal torsion coset there could possibly be, and then to give a method for finding them all.