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The reference below shows if $n!$ is ultimately hard, then $P_{\mathbb{C}} \ne NP_{\mathbb{C}}$. Is there any implication about the case if $n!$ is easy?

On the intractability of Hilbert's Nullstellensatz and an algebraic version of "$NP \ne P$" (1996) by Michael Shub , Steve Smale.

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What do you mean by the complexity of $n!$? Do you mean the complexity of computing $n!$? What do you mean by $P_{\mathbb{C}}$? –  Qiaochu Yuan Jul 29 '12 at 17:29
    
I am considering the model by Shub and Smale ("On the intractability of HIlbert's Nullstellansatz and an algebraic version of "$NP \ne P$""). –  J.A Jul 29 '12 at 17:31
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That would have been a useful thing to mention in your original question. Shub and Smale paper available here. –  MJD Jul 29 '12 at 17:46
    
There's no reverse implication, in that paper at least. The computation time of the sequence $n!$ only gives a lower bound for the Twenty Questions problem they set out. –  Luke Mathieson Jul 30 '12 at 1:44
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