关于易处理指数和
On Tractable Exponential Sums
摘要：
我们考虑的问题，评估某些指数和。这些和采用$\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 πi / N}} $的形式，其中每个x_i在环Z_N上求和，并且f（x_1，.，x_n）是具有整数系数的多元多项式。我们表明，和可以评估在多项式时间在n和log N时，f是一个二次多项式。即使N的因式分解未知，这也是正确的。以前，这是已知的素数模N。另一方面，对于非常具体的家庭的多项式的程度\ge 3，我们证明了这个问题是P-硬，即使是任何固定的素数或素数功率模。这导致了一个复杂性二分法定理--对于一类指数和，每个问题要么可以在多项式时间内计算，要么可以在#P-困难时间内计算。这些和出现在图同态的分类和其他一些计数CSP类型的问题，这些结果导致复杂性二分法定理。对于多项式时间算法，高斯和形成基本构建块。对于硬度的结果，我们证明组理论的必要条件，易处理。这些测试意味着问题是P-困难的，即使是非常有限的家庭简单的三次多项式在固定模N。
英文摘要：
We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 πi / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N.
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