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

TL;DR

This work proves an exponential-time lower bound for the QUAD feasibility problem over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Z}$ within the BCSS framework, thereby establishing $P_k \neq NP_k$ for these domains. It extends the scheme-theoretic approach from Civril21 by constructing prime homogeneous simple sub-problems to amplify instance counts and derive tight lower bounds. The key technique combines a constructive induction that yields $2^r$ instances on $3r$ variables plus a cycle-based primeness enforcement, aided by a Stirling-based bound to show $\kappa(\mathrm{QUAD}) \ge 2^{(\frac{1}{3}-\epsilon)n}$ with $n=3r$. Coupled with the NP-completeness of QUAD, the results yield $P_{\mathbb{C}} \neq NP_{\mathbb{C}}$, $P_{\mathbb{R}} \neq NP_{\mathbb{R}}$, and $P_{\mathbb{Z}} \neq NP_{\mathbb{Z}}$, marking explicit separations in the BCSS model for these fields/rings.

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.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2: Fundamental Lemma
• Theorem 3
• proof
• Corollary 4
• Corollary 5
• Corollary 6