Planarity of products of two linearized polynomials

Let L1(x) and L2(x) be linearized polynomials over Fqn. We give conditions when the product L1(x)⋅L2(x) defines a planar mapping on Fqn. For a polynomial L over Fqn, let M(L)={α∈Fqn:L(x)+α⋅xis bijective onFqn}. We show that the planarity of the product L1(x)⋅L2(x) is linked with the set M(L) of a suitable linearized polynomial L. We use this relation to describe families of such planar mappings as well as to obtain nonexistence results.