齐次Keller映射的碰撞-外壳压缩与赵消失猜想的四十变量反例
Collision-Generated Compression for Homogeneous Keller Maps
- Queen Mary University of London（伦敦大学玛丽女王学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出齐次Keller映射的碰撞生成压缩原理，利用极化代数构造子代数，基于Thompson的24变量映射得到20维子空间，构造出赵消失猜想的四十变量反例并验证相关计算。
AI中文摘要：
我们提出了一种针对齐次Keller映射的初等碰撞生成压缩原理。在极化代数的框架下，该构造取由一次碰撞生成的子代数；额外的要点在于，该子代数带有非单射的Keller限制，并控制着标准对称提升的维数。对于Thompson的24变量三次齐次映射，精确极化给出增长序列2、4、11、20、20，且所得子代数恰好是MacFarlane的20维不变子空间。由此，已知的24到20的压缩可直接从碰撞本身正则地得到。尽管所得的40变量提升已被提及，我们仍给出了一个显式的含350个单项式的齐次四次多项式，使得赵消失猜想在此处失效，同时还给出了在Q(i)上的精确梯度碰撞。在Q、Q(i)和C上，在未改变的对称提升之前，不存在包含该碰撞的真不变线性子空间。我们不主张任何全局极小性。主要的显式计算由配套的精确计算验证。
英文摘要：
We associate to a collision $p\ne q$ of a homogeneous Keller map $F=\operatorname{id}+h$ 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 $\mathbb{Q}(i)$. The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.
