# The permanent conjecture

 Importance: Medium ✭✭
 Author(s): Kahn, Jeff
 Subject: Combinatorics » Matrices
 Keywords: invertible matrix permanent
 Recomm. for undergrads: no
 Prize: none
 Posted by: mdevos on: March 8th, 2007
Conjecture   If is an invertible matrix, then there is an submatrix of so that is nonzero.

If true, this conjecture would imply the nowhere-zero point in a linear mapping conjecture via the Alon-Tarsi polynomial technique. I believe Yang Yu was the first to suggest the following generalization of the permanent conjecture.

Conjecture  (Yu)   If are invertible matrices over the same field, then there is an submatrix of so that is nonzero.

This conjecture when restricted to the field is a consequence of the Alon-Tarsi basis conjecture. In addition to implying the above conjecture, the truth of this conjecture for matrices over the field would imply that every 6-edge-connected graph has a nowhere-zero 3-flow, thus resolving The weak 3-flow conjecture.