Collision-Generated Compression for Homogeneous Keller Maps

Abstract

We formulate a collision-generated compression principle for homogeneous Keller maps. In the language of polarization algebras, the construction takes the subalgebra generated by a collision; the additional point is that this subalgebra carries a noninjective Keller restriction and controls the dimension of the standard symmetric lift. We give two exact applications. For Thompson's 24-variable cubic-homogeneous map, the polarization-subalgebra growth is \(2,4,11,20,20\), and the resulting subalgebra is exactly MacFarlane's 20-dimensional invariant subspace, thereby recovering the known \(24\)-to-\(20\) compression canonically from the collision. For Van Rijn's subsequent 19-variable cubic-homogeneous map, arising from a 12-variable degree-three Keller map, the corresponding growth is \(2,4,11,19,19\), so the known collision generates the whole space. Thus no proper invariant linear restriction retaining that collision can improve its 38-variable symmetric lift. We make this lift explicit: the resulting homogeneous quartic has 340 monomials, is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over \(\Q(i)\). For comparison, the 20-variable application gives the previously studied 40-variable, 350-monomial quartic. No global minimality is claimed. All finite calculations are checked by accompanying exact code.