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