What is the $\Omega$ complexity of $n^2*m+10*n*m^2$ in terms of $m$?

Question

I have an algorithm that runs in time $n^2*m+10*n*m^2$.

Assuming that $n> m^2$ What is its "big-omega" complexity in terms of $m$? Isn't the big omega for this $m$?

Answer 1

With the assumption $n > m^2$, the best you can say is $\Omega(m^5)$. (Of course you can say looser things like $\Omega(m)$ or $\Omega(1)$ too.)

$$n^2 m + 10nm^2 > m^5 + 10 m^4 \ge 2 m^5,\qquad \text{for all large $m$}$$ where the last step uses the fact that $m^5 \ge 10m^4$ for all large $m$ (say, $m \ge 10$).