TITLE: asymptotically larger vs polynomially larger
QUESTION [9 upvotes]: What is the difference between x being asymptotically larger than y and x being polynomially larger than y?

REPLY [7 votes]: I don't agree with the definition of polynomially larger in the other answer, so here's the one I see used:
"Polynomially larger" means that the ratio of the functions falls between two polynomials, asymptotically. Specifically, $f(n)$ is polynomially greater than $g(n)$ if and only if there exist generalized polynomials (fractional exponents are allowed) $p(n),q(n)$ such that the following inequality holds asymptotically:
$$p(n)\leq \frac{f(n)}{g(n)}\leq q(n)$$
This is more strict than merely asymptotically larger, and gives limits on how much faster the function can grow. For example, $n$ is not polynomially larger than $\frac{n}{\ln(n)}$ because the rate of growth is wrong. To see this, notice that there's no polynomial $p$ such that $$p(n)\leq\frac{n}{\frac{n}{\ln(n)}}=\ln(n)$$ holds asymptotically.
Another example of this is $e^n$, which is not polynomially larger than $n^k$ for any $k$. In both of these examples, the correct interpertation is that the function is asymptotically larger, but not polynomially so on the grounds of growing too quickly compared to $f(n)=n^k$ for the gap to be characterized as "polynomial." $e^n$ is exponentially larger than $n^k$, where "exponentially larger" is defined similarly to "polynomially larger" except using exponentials instead of polynomials.