M.Sc Thesis | |

M.Sc Student | Karelin Ariel |
---|---|

Subject | Bounded Generation in SLn(Fq{X}) |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Chen Meiri |

Full Thesis text |

Gaussian elimination works over fields. This fact, taught
in any first year linear algebra course, states that over any field *F*,
every square matrix of order *n *and determinant *1* over *F* is
a product of *O(n ^{2})* elementary matrices, i.e. matrices having

When moving to matrices of determinant *1*
over Euclidean domains, there is still an algorithm resembling Gaussian
elimination, transforming every matrix of determinant 1 to the identity after a
finite number of steps. However, this method makes a repeated use of Euclid's
algorithm, and hence the number of elementary operations needed for the
elimination may be unbounded. We say that elementary matrices generate *SL _{n}*
over Euclidean domains, yet perhaps unboundedly. When generalizing to more
general commutative rings, even generation itself cannot be guaranteed.

Given a ring *R* (which is not a field)
and a nonzero ideal *J*, a *J*-elementary matrix is an elementary
matrix whose nonzero off-diagonal entry belongs to *J*. A sensible generalization
of the field case asks whether *J*-elementary matrices boundedly generate
the subgroup they generate. We answer this question affirmatively for *R=F _{q}[x]*,
where

We also present two applications of our result on bounded generation:

Word width in *SL _{n}(F_{q}[x])*:
Let

Ulam Stability of *SL _{n}(F_{q}[x])*:
A group