Scheme-theoretic Approach to Computational Complexity II. The Separation of P and NP over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Z}$

Abstract

We show that the problem of determining the feasibility of quadratic systems over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Z}$ requires exponential time. This separates P and NP over these fields/rings in the BCSS model of computation.