维数大于二的雅可比猜想的反例
Counterexamples to the Jacobian conjecture in dimensions greater than two
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中文总结 AI 辅助
该研究将雅可比猜想的反例从三维推广到所有大于二的维数，构造了多个不同维数、不同次数的显式非平展覆盖反例，验证了其正确性并确定了纤维结构。
中文摘要 AI 辅助
雅可比猜想自1939年以来一直悬而未决，它询问：所有从$\boldsymbol{\textrm{C}}^n$到自身的多项式映射，若其雅可比行列式为非零常数，是否一定存在多项式逆映射。2026年7月19日，Alpöge在三维维数下否定了该猜想；7月20日，Gallagher给出了无限族反例；7月23日，Speyer给出了几何解释：该反例扫过平面曲线的切线，而经典对偶性迫使这种映射多次击中大部分点。我们对这种切线扫描机制给出了自包含的阐述，并将其从平面曲线推广到超曲面上的方向场。所得构造在每个维数大于二的维数中都产生反例，且在每个维数中都具有任意大的几何次数（典型点的原像个数）。我们给出了五个新的显式映射：一个三维的次数为四，两个四维的次数分别为五和十，两个五维的次数分别为六和十二。这些反例提供了非真的平展覆盖$\boldsymbol{\textrm{C}}^n\to\boldsymbol{\textrm{C}}^n$的显式例子：它们处处非分歧，仅在点逃逸至无穷时不具单射性。所有恒等式均通过精确有理算术验证；附录通过格罗比纳基确定了精确的纤维结构。
英文摘要
The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\C^n$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve --- a map that classical duality forces to hit most points several times. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five new explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. The counterexamples provide explicit examples of étale coverings $\C^n\to\C^n$ that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic; an appendix determines exact fiber structures through Gröbner bases.
