= Polynomial kernel
{title2=$k(x,z)=(x^Tz+c)^r$}
{wiki}
For $x,z\in\mathbb R^d$, a polynomial kernel is the <positive-semidefinite kernel> $k(x,z)=(x^Tz+c)^r$, where $r$ is a positive integer and $c\geq0$. The case $c=0$ uses <monomials> of total degree $r$; $c>0$ also includes lower degrees. The <feature map> weights matter: listing each distinct <monomial> once with coefficient one generally gives a different <inner product>.<Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams>[Section 4.2.2, pp. 89–90, equations (4.23)–(4.24): polynomial kernels and weighted monomial features]{ref}
For the homogeneous quadratic case in two dimensions, take the scaled <quadratic feature map>
$$
\phi(x_1,x_2)=\begin{pmatrix}x_1^2\\\sqrt2 x_1x_2\\x_2^2\end{pmatrix},\qquad k(x,z)=(x^Tz)^2.
$$
The <inner product> identity is explicit:
$$
\phi(x)^T\phi(z)=x_1^2z_1^2+2x_1x_2z_1z_2+x_2^2z_2^2=(x_1z_1+x_2z_2)^2=k(x,z).
$$
The factor $\sqrt2$ accounts for the two equal cross terms in the square. Thus an <inner product> of three derived coordinates can be obtained from an <inner product> of the two input coordinates followed by squaring. The <kernel trick> uses this equality inside an algorithm; the <feature map> explains which nonlinear <functions> the algorithm can represent.
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