Matrix Choosability

Let F be a finite field with pc elements, let A be a n×n matrix over F , and let k be a positive integer. When is it true that for all X1, . . . , Xn ⊆ F with |Xi| = k+1 and for all Y1, . . . , Yn ⊆ F with |Yi| = k, there exist x ∈ X1×. . .×Xn and y ∈ (F \Y1)×. . .×(F \Yn) such that Ax = y? It is trivial that A has this property for k = pc − 1 if det(A) 6= 0. The permanent lemma of Noga Alon proves that if perm(A) 6= 0, then A has this property for k = 1. We will present a theorem which generalizes both of these facts, and then we will apply our theorem to prove “choosability” generalizations of Jaeger’s 4-flow and 8-flow theorems in Zk p .