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)≠0. The permanent lemma of Noga Alon proves that if perm(A)≠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 Jaegerʹs 4-flow and 8-flow theorems in Zkp.