On hardness of multilinearization, and VNP-completeness in characteristics two

For a boolean function f : {0, 1} → {0, 1}, let f̂ be the unique multilinear polynomial such that f(x) = f̂(x) holds for every x ∈ {0, 1}. We show that, assuming VP 6= VNP, there exists a polynomialtime computable f such that f̂ requires super-polynomial arithmetic circuits. In fact, this f can be taken as a monotone 2-CNF, or a product of affine functions. This holds over any field. In order to prove the results in characteristics two, we design new VNPcomplete families in this characteristics. This includes the polynomial ECn counting edge covers in a graph, and the polynomial mcliquen counting cliques in a graph with deleted perfect matching. They both correspond to polynomial-time decidable problems, a phenomenon previously encountered only in characteristics 6= 2.