陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「A digestion of the Jacobian conjecture counterexample」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem , but is omitted here; see also Lemma 5 of this previous blog post .) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Ja

已知结果和反例

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

证明或构造的主线

The example can be stated completely explicitly: one can take

The example has since been retroactively explained in more geometric terms . As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

阅读时建议盯住的点

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, 下面会 show

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

值得单独记下的条目

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.
• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .
• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源