Polynomial is of Exponential Order Epsilon

Theorem

Let $P: \R \to \mathbb F$ be a polynomial, where $\mathbb F \in \set {\R, \C}$.

Then $P$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.

Proof

If $P = 0$, the theorem holds trivially.

Let $P_n$ be a polynomial of degree $n$, where $n \ge 0$.

The proof proceeds by induction on $n$, where $n$ is the degree of the polynomial.

Basis for the Induction

Let $P_0$ be of degree zero.

Then $P_0$ is a constant polynomial.

By Constant Function is of Exponential Order Zero, $P_0 \in \mathcal E_0$.

Therefore, by Raising Exponential Order, it is of exponential order $\epsilon$ as well.

This is the basis for the induction.

Induction Hypothesis

Fix $n \in \N$ with $n \ge 0$.

Assume:

$P_n \in \mathcal E_\epsilon$

That is:

$\size {\map {P_n} t} < K e^{\epsilon t}$

for some $K > 0$, for $\epsilon > 0$ arbitrarily small.

This is our induction hypothesis.

Induction Step

Let $P_{n + 1}$ be of degree $n + 1$.

By the definition of polynomial,

$P_{n + 1} = P_n + a_{n + 1} x^{n + 1}$

for some polynomials of degree $n$.

$P_n$ is of exponential order $\epsilon$ by the induction hypothesis.

Thus, by:

Sum of Functions of Exponential Order
Scalar Multiple of Function of Exponential Order
Natural Number Power is of Exponential Order Epsilon

We have that $P_{n + 1}$ is of degree $\epsilon$.

The result follows by the Principle of Mathematical Induction.

$\blacksquare$