Alon–Tarsi numbers of direct products

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), v\in V$, and the graph polynomial $\prod_{(i,j)\in E} (x_j-x_i)$ contains an ``almost central'' monomial (that means a monomial $\prod_v x_v^{c_v}$, where $|c_v-d(v)|\leqslant 1$ for all $v\in V$), then the Cartesian product $G\square C_{2n}$ is $(d(\cdot)+2)$-choosable.