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「Distinguished Lecture Series III: Avi Wigderson, “Algebraic computation”」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

Avi Wigderson ‘s final talk in his Distinguished Lecture Series on “ Computational complexity ” was entitled “ Arithmetic computation “; the complexity theory of arithmetic circuits rather than boolean circuits .

Arithmetic circuits manipulate inputs and outputs in a given field, such as the complex numbers (or more generally, a field of characteristic 0, to avoid having to deal with arithmetic “shortcuts” coming from identities such as in ). There are many models for such circuits, but one simple one is to create circuits by composing addition gates (which take two inputs x and y and return x+y as an output), multiplication gates (which take two inputs x, y and return xy as an output

已知结果和反例

For simplicity let us view the scalar multiplication gates as being “free” and only count the addition and multiplication gates when counting complexity. Thus, for instance, to perform an inner product computation

requires 2n-1 gates (n to multiply the pairs and together, and then n-1 to add together the resulting pairs). It is not difficult to show that this 2n-1 is sharp. More generally, given any polynomial operation p with many inputs and one output, we define the circuit complexity S(p) of p to be the least number of gates needed to build an arithmetic circuit that can generate p from ; it is clear that p has to be polynomial in order to be generated by such circuits. We can also

证明或构造的主线

More generally, the problem of computing S(p) or for various interesting polynomials is an important one, to which there are still only very partial answers. Even the complexity of a single polynomial p(x) of a single variable x is not easy. If p has degree d, then clearly , and counting arguments can show that this is sharp for “generic” p, but for many p one can do a lot better; for instance it is not hard to see that by repeatedly squaring x and then combining some of the

Avi noted, though, that it was still an open question to find an “natural” polynomial p of degree d for which S(p) grew faster than any power of . (One could take a polynomial p whose coefficients are all algebraically independent, but this is “cheating”.) In particular, it is not known if

阅读时建议盯住的点

for every C. But… if this claim failed, then one can show that factoring n-bit numbers can be done in polynomial time! (The reason is that if (1) failed, then one can compute d! mod N quickly for n-bit integers d and N by using a polynomial size circuit modulo N, which basically allows one to determine whether all prime factors of N are less than any given threshold d, from which one can quickly isolate each individual prime factor.) But this already shows how difficult we ex

Here are some other interesting examples of polynomials whose complexity we would like to understand:

值得单独记下的条目

• The symmetric polynomial of degree d on n variables. By recursively describing the symmetric polynomials of degree up to d in terms of the same polynomials on one fewer variable, one can obtain an upper bound . The trivial lower bound is .

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