Ultimate Polynomial Time
Gregorio Malajovich
TL;DR
The paper defines the ultimate polynomial-time class $\mathcal{UP}$ for decision problems over $\mathbb{C}$ within the Blum–Shub–Smale framework, leveraging definability without constants and Zariski-geometric decompositions. It connects the $\tau$-conjecture to lower bounds and to the separation $\mathcal{P} \neq \mathcal{NP}$ over $\mathbb{C}$, showing $\mathcal{P} \cap \mathcal{K} \subseteq \mathcal{UP}$ and establishing an NP-hardness boundary for $\mathcal{UP}$ via the Hilbert Nullstellensatz. The core results demonstrate a chain of implications among conjectures and provide a constructive method to associate input-dependent polynomials $f_i$ (with $\tau(f_i)$ poly-bounded) to the canonical paths on irreducible components, yielding a framework for ultimate lower bounds on structured problems. The notion of ultimate complexity is further developed to quantify worst-case and asymptotic lower bounds (e.g., ultimate logarithmic/exponential time) for such problems, offering a new lens for proving hardness in algebraic computation.
Abstract
The class $\mathcal{UP}$ of `ultimate polynomial time' problems over $\mathbb C$ is introduced; it contains the class $\mathcal P$ of polynomial time problems over $\mathbb C$. The $τ$-Conjecture for polynomials implies that $\mathcal{UP}$ does not contain the class of non-deterministic polynomial time problems definable without constants over $\mathbb C$. This latest statement implies that $\mathcal P \ne \mathcal{NP}$ over $\mathbb C$. A notion of `ultimate complexity' of a problem is suggested. It provides lower bounds for the complexity of structured problems.