Question
I want to determine the order of growth of a complex polynomial $p(z)$.
My attempt
We pick $$p(z)=a_nz^n+\ldots +a_1z+a_0$$
then we know that
$$|p(z)| \le |a_nz^n|+\ldots +|a_1z|+|a_0|$$
since $e^x$ is increasing we have that
$$|p(z)|<e^{|p(z)|} \le e^{|a_nz^n|+\ldots +|a_1z|+|a_0|}=e^{|a_n||z^n|+\ldots +|a_1||z|+|a_0|} \le max\{|a_i|\}e^{max\{|a_i|\}|z|^{n}}$$
The thing is that for small values of $z$ for example $z=0$ the above bound doesn't work and my definition says that the above should hold for all $z \in \mathbb{C}$. I am almost sure that the order of growth should be $n$ but I don't know if this is correct or if there is a better proof to get which is the order of growth of a complex polynomial that you can provide.
Definitions.
Let $f$ be an entire function. If there exist a positive number $\rho$ and constants $A,B >0$ such that
$$|f(z)| \le A e^{B|z|^{\rho}}$$
for all $z \in \mathbb{C}$ then we say that $f$ has order of growth $\le \rho$. We define the order of growth of $f$ as
$$\rho_f=inf\{ \rho \}$$
Best Answer
For all real $r$ we have $e^r\ge 1+r$, hence for positive $r$ we have $$e^r=(e^{r/n})^2\ge (1+r/n)^n\ge 1+\frac1{n^n}\cdot r^n $$ or in other words with $\rho=\frac1n$ and $r=(n\left|z\right|)^\rho$, we have $$1+|z|\le e^{n^\rho|z|^\rho} $$ and so for any a linear polynomial $p(z)=az+b$ we find that for $n\in\Bbb N$, $\rho=\frac 1n$, $B=n^\rho$, $A=\max\{|a|,|b|\} $ $$ |az+b|\le|a|\cdot|z|+|b|\le A(1+|z|)\le Ae^{B|z|^\rho}.$$ As this works for any $n\in\Bbb N$, we conclude that $az+b$ has order of growth $0$. The key observation is that even the slightest exponetial grows faster than linear. (It also grows faster than any polynomial, bit as we have the fundamantal theorem of algebra available, we can reach the result for arbitrary polynomials by a helpful result that is interesting in its own right:
Next note that the growth rate of the product of two functions is at most the max of their growth rates. Indeed, if $|f(z)|\le Ae^{B|z|^\rho}$ and $|g(z)|\le A'e^{B'|z|^{\rho'}}$ then $$|f(z)g(z)|\le AA'e^{B|z|^\rho+B'|z|^{\rho'}} $$ and as $$B|z|^\rho+B'|z|^{\rho'}\le (B+B')|z|^{\max\{\rho,\rho'\}} +\max\{B,B'\}$$ we find $$|f(z)g(z)|\le AA'e^{\max\{B,B'\}}\cdot e^{(B+B')|z|^{\max\{\rho,\rho'\}}}. $$
So finally, as every polynomial is the product of linear factors, we conclude that every polynomial has order of growth $0$.