Project: Research

- Smyth, Chris (Principal Investigator)
- Aliev, Iskander (Researcher)

Status | Finished |
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Effective start/end date | 18/01/05 → 30/11/08 |

Total award | £154,359.00 |
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Funding organisation | EPSRC |

Funder project reference | GR/T04458/01 |

Period | 18/01/05 → 30/11/08 |
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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.

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.

## Solving algebraic equations in roots of unity

Research output: Contribution to journal › Article

## Siegel's lemma and sum-distinct sets

Research output: Contribution to journal › Article

## An optimal lower bound for the Frobenius problem

Research output: Contribution to journal › Article

## Successive minima and best simultaneous Diophantine approximations

Research output: Contribution to journal › Article

## Best simultaneous Diophantine approximations under a constraint on the denominator

Research output: Contribution to journal › Article

## Lattice points in large Borel sets and successive minima

Research output: Contribution to journal › Article

## On the lines passing through two conjugates of a Salem number

Research output: Contribution to journal › Article

## Mahler measure of one-variable polynomials: a survey.

Research output: Chapter in Book/Report/Conference proceeding › Chapter (peer-reviewed)

## Finding maximal torsion cosets on varieties

Research output: Contribution to conference › Paper