Collision-Generated Compression for Homogeneous Keller Maps
Abstract
We associate to a collision p≠ q of a homogeneous Keller map F=id+h the polarization subalgebra generated by p and q. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over ℚ(i). The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.
Related
- Keller maps and holonomic D-modules
- Quartic Keller Maps and Tameness in Three Variables
- A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
- Keller's conjecture for split finite-dimensional algebras
- Hybridizable Discontinuous Galerkin methods for the Keller-Segel model
BibTeX
@misc{prellberg2026collisiongenerated,
title = {{Collision-Generated Compression for Homogeneous Keller Maps}},
author = {Thomas Prellberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12543},
eprint = {2608.12543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12543},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Collision-Generated Compression for Homogeneous Keller Maps
AU - Thomas Prellberg
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.12543
UR - https://arxiv.org/abs/2608.12543
AB - We associate to a collision p≠ q of a homogeneous Keller map F=id+h the polarization subalgebra generated by p and q. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over ℚ(i). The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.
ER -