The Alon-Tarsi basis conjecture
Definition: If is an matrix over a field of characteristic , then we say that is an Alon-Tarsi basis (or AT-basis) if the permanent of the matrix obtained by stacking copies of is nonzero.
It follows from the Alon-Tarsi polynomial technique that if is an AT-base then for every of size 2 and for every , there exists a vector so that (using the notation from A nowhere-zero point in a linear mapping, is (2,1)-choosable). It follows from this that every Alon-Tarsi base over is also an additive basis. Thus, the above conjecture, if true, would imply The additive basis conjecture. The following strengthening of this conjecture was suggested in [D]