Vortrag
Polynomial Matrices over Finite Fields with Given Smith Form
Vortrag von Prof. Dr. Samrith Ram
Datum: 20.12.17 Zeit: 12.00 - 13.00 Raum:
Let m; n be positive integers and denote by F_q the finite field with
q elements. Let V be a vector space of dimension mn over F_q and
T : V --> V be a linear transformation. An m-dimensional subspace W
of V is said to be T-splitting if
V = W + TW + ... + T^(n-1)W:
Determining the number of m-dimensional T-splitting subspaces for an
arbitrary transformation T is an open problem closely related to many
other problems in combinatorics. I will outline some of them including
connections with a theorem of Philip Hall on conjugacy class size in
the general linear group and some results of Wilf and others on the
probability of coprime polynomials over finite fields. I will also discuss
a general counting problem involving polynomial matrices which, if
solved, would settle the problem of counting T-splitting subspaces.
1