The Radical Solution and Computational Complexity
Bojin Zheng, Weiwu Wang
TL;DR
The paper investigates the complexity status of radical solutions to polynomials with rational coefficients within the $P$ vs $NP$ framework. It introduces a running-graph and reversibility-based approach, arguing that a hypothetical $P$-algorithm induces a one-way graph $G$ while a corresponding $NP$-algorithm yields a two-way graph $G'$ isomorphic to $G$ augmented by its reverse $G^{-1}$, thereby tying radical formulas to a fundamental limitation. It argues that solving radical roots over $\mathbb{Q}$ is in $\mathbb{NP}$, yet no general deterministic polynomial-time algorithm exists for higher-degree polynomials, supporting $P \neq NP$ and framing this as an impossible trinity among generality, determinism, and efficiency. The work further discusses representations of roots (radical, exponential, direct) and the role of reversible computation, offering a novel lens on why a universal radical formula cannot exist. Overall, the paper blends algebraic solvability with complexity theory to propose a framework linking radical solvability to fundamental limits of efficient computation.
Abstract
The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $\mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ \mathscr{P} \in \mathbb{P}$ shall have a one-way running graph $G$, and have a corresponding $\mathscr{Q} \in \mathbb{NP}$ which have a two-way running graph $G'$, $G$ and $G'$ is isomorphic, i.e., $G'$ is combined by $G$ and its reverse $G^{-1}$. When $\mathscr{P}$ is an algorithm for solving polynomials, $G^{-1}$ is the radical formula. According to Galois' Theory, a general radical formula does not exist. Therefore, there exists an $\mathbb{NP}$, which does not have a general, deterministic and polynomial time-complexity algorithm, i.e., $\mathbb{P} \neq \mathbb{NP}$. Moreover, this paper pointed out that this theorem actually is an impossible trinity.