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What is the difference between x being asymptotically larger than y and x being polynomially larger than y?

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Here's one possible interpretation.

We say that $x$ is asymptotically larger than $y$ if $\lim_{n \rightarrow \infty} x(n)/y(n) = \infty$. We write it $x \gg y$ or $y \ll x$ or $x = \omega(y)$ (rarely) or $y = o(x)$.

We say that $x$ is (at most) polynomially larger than $y$ if there is some $d$ such that $x = O(y^d)$, that is $\lim_{n \rightarrow \infty} x(n)/y(n)^d < \infty$, in other words there is some constant $C$ such that $x(n) \leq Cy(n)^d$.

Note that there is a very confusing semantic difference: the first notion is "strict" (like $>$), the second notion is "lax" (like $\leq$).

It's possible that different areas of mathematics understand these notions slightly differently.

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