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

TL;DR

This paper demonstrates that deciding the feasibility of quadratic systems over complex numbers, real numbers, and integers requires exponential time, establishing a separation between P and NP over these fields in the BCSS model.

Contribution

It provides the first known exponential time lower bounds for quadratic feasibility problems over these fields, separating P and NP in the algebraic setting.

Findings

Quadratic feasibility over $\

$ ext{, } ext{, and } ext{ requires exponential time.

Establishes P ≠ NP over $ ext{, } ext{, and } ext{ in the BCSS model.

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.