Alon–Tarsi for Hypergraphs

Given a hypergraph $H=(V, E)$, define for every edge e∈ E a linear expression with arguments corresponding to the vertices. Next, let the polynomial pH be the product of such linear expressions for all edges. Our main goal is to find a relationship between the Alon-Tarsi number of pH and the edge density of H. We prove that AT(pH)= ed(H)+1 if all the coefficients in pH are equal to $1$ and the base field has characteristic zero. Our main result is that, over an arbitrary field, if on every edge the coefficients are not all equal, then they can be permuted within the edges so that for the resulting polynomial pH^, AT(pH⁾≤ 2 ed(H)+1 holds. We conjecture that this bound holds for every hypergraph polynomial without permuting its coefficients. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.