Ultimate Polynomial Time
Gregorio Malajovich
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 $\tau$-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.