AI 中文总结
该研究证明弱马库斯-山下猜想在n≥14维不成立，通过链实现定理构造出对应Hurwitz向量场，给出14维和18维的具体反例，3至13维的情况尚未解决。
AI 中文摘要
我们证明弱马库斯-山下猜想在所有维度n≥14时均不成立。我们首先证明一个链实现定理：对于ℝⁿ上的每个多项式Keller映射F=I+H，其分量次数为d₁,…,dₙ，可在ℝᴺ上构造显式多项式向量场，其中N=Σᵢmax(dᵢ,2)-n，该向量场的雅可比矩阵在每一点的谱均为{-1}，且其奇点与F的任意指定纤维一一对应。将此定理应用于近期雅可比猜想的反例，可得到ℝ¹⁴上的7次Hurwitz向量场，具有3个有理奇点；对相关三次稳定化进行秩降低的去齐次化，还可独立得到ℝ¹⁸上的3次Hurwitz场。维度3≤n≤13的情况仍待解决。
英文摘要
We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.
