Recently I am reading a textbook which discusses about kernel function. It says:
if $p: X \to \mathbb R$ is a polynomial of the form $p(t)=a_m t^m+ \cdots + a_1 t + a_0$ with non-negative coefficients $a_i$, then $k(x,x'):= p(xx'), x, x' \in X$, defines a kernel on $X$. In general, computing this kernel needs its feature map $\Phi(x):= (\sqrt a_m x^m ,..., \sqrt a_1 x, \sqrt a_0 ), x \in X$, and consequently the computational requirements are determined by the degree $m$. However, for some polynomials, these requirements can be substantially reduced. Indeed, if for example we have $p(t)=(t+c)^m$ for some $c>0$ and all $t \in \mathbb R$, then the time needed to compute $k$ is independent of m.
Question: I don't understand that if $p(t)=(t+c)^m$ , then the time needed to compute $k$ is independent of m.