On Tractable Exponential Sums

TL;DR

This paper classifies the computational complexity of evaluating exponential sums with polynomial functions, showing polynomial-time algorithms for quadratic cases and #P-hardness for higher degrees, establishing a complexity dichotomy.

Contribution

It provides a complete classification of the complexity of exponential sums based on polynomial degree, extending known results to composite moduli, and introduces group-theoretic criteria for hardness.

Findings

Quadratic exponential sums can be evaluated in polynomial time even for composite moduli.

Higher-degree polynomials (degree ≥ 3) lead to #P-hard problems, even for simple cases.

A complexity dichotomy theorem is established, classifying problems as either tractable or #P-hard.

Abstract

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 \pi i / 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.