Proof of the oval conjecture for planar partition functions

We prove that the translation plane and the shift plane defined by a planar partition function form an oval pair of projective planes in the sense that the planes share a line pencil and any line of either plane not in this pencil forms an oval in the other plane. This is achieved by building upon substantial work of Betten–L¨owen and by using Rabier’s fibration theorem which allows to conclude — without the assumption of properness — that certain local diffeomorphisms are covering maps. A function f : Rn → Rn is called planar, if the map fd : Rn → Rn , x → f(x + d) − f(x) is bijective for all d ∈ Rn \ {0}. It is called a partial spread function, if it is continuously differentiable, the derivative Df : Rn → L(Rn , Rn ) is injective and the set Sf := {{(ξ, Df(x) · ξ) : ξ ∈ Rn } : x ∈ Rn } of n-dimensional subspaces of Rn ×Rn is a partial spread, i.e. any two subspaces intersect in the trivial subspace; if Sf ∪ {{0} × Rn } is a spread, i.e. in addit...

• Type: Journal
• Year: 2007
• Where: EJC
• Authors: Nils Rosehr
• Keywords: EJC 2007, Information Technology, Partition Function, Planar Partition Function, Projective Planes