AI 中文总结
研究\(n\times n\)矩阵行列式计算，将赖瑟和格林算法重铸为含正交性准则的单一原则，指出离散傅里叶变换等是其实例，分析各方案特点，如离散傅里叶变换方案的求值方式、运行操作及相关条件，还提及哈达玛和循环方案的特性及满足条件的最小长度。
AI 中文摘要
\(n\times n\)矩阵的行列式是其行形式\(\prod_i\bigl(\sum_j b_{ij} x_j\bigr)\)乘积中完全混合单项式\(x_0 \cdots x_{n - 1}\)的系数，赖瑟和格林的经典精确算法通过在布尔立方体上对该乘积进行\(2^{n - 1}\)次求值求和来计算它。我们将此重铸为一个单一原则——一种求值方案以及一个正交性准则，该准则能直接判定其加权和是否等于行列式，并表明赖瑟公式、格林公式和离散傅里叶变换是它的三个实例：\(0 - 1\)和\(\pm 1\)哈达玛系统，以及长度为\(N\) 的单个循环变换。离散傅里叶变换方案将行列式\(\mathrm{per}\,B\)评估为模\(x^N - 1\)的单变量多项式的一个系数；当指数集对模\(N\)有效时它是精确的，运行操作与经典公式相同为\(\Theta(2^n n)\)，在有限域上是数论变换，通过中国剩余定理返回精确的整数行列式。哈达玛方案免费具有正交性，循环方案则以指数上的算术条件来交换；在配套论文arXiv:2607.08366中确定其能满足的最小长度为\(N = 2^n - 2^{\lfloor \log_2 n \rfloor}\)。所有这三种都是有限阿贝尔群上的特征和，并且该准则恰好适用于此类群。
英文摘要
The permanent of an $n \times n$ matrix is the coefficient of the fully mixed monomial $x_0 \cdots x_{n-1}$ in the product of its row forms $\prod_i\bigl(\sum_j b_{ij} x_j\bigr)$, and the classical exact algorithms of Ryser and Glynn compute it by summing $2^{n-1}$ evaluations of that product over the Boolean cube. We recast this as a single principle -- an evaluation scheme together with an orthogonality criterion that decides, in one line, whether its weighted sum equals the permanent -- and show that Ryser's formula, Glynn's formula, and a discrete Fourier transform are three instances of it: the $0$--$1$ and $\pm 1$ Hadamard systems, and a single cyclic transform of length $N$. The DFT scheme evaluates $\mathrm{per}\,B$ as one coefficient of a univariate polynomial modulo $x^N - 1$; it is exact exactly when the exponent set is valid mod $N$, runs in the same $Θ(2^n n)$ operations as the classical formulas, and over a finite field is a number-theoretic transform that returns the exact integer permanent by the Chinese remainder theorem. Where the Hadamard schemes are orthogonal for free, the cyclic scheme trades this for an arithmetic condition on the exponents; the smallest length at which it can be met is $N = 2^n - 2^{\lfloor \log_2 n \rfloor}$, established in the companion paper arXiv:2607.08366. All three are character sums over a finite abelian group, and the criterion holds precisely for such groups.
Commentsv2: added Corollary 1 (partial-derivative lower bound for evaluation schemes and abelian groups); 10 pages
