Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as:
$$ K: x,y \mapsto (x^Ty + c)^2, $$ where $c\ge 0$. What happens if $c < 0$? Take the following kernels for example:
$$ \forall x,y \in \mathbb{R} \begin{cases} K_+: (x,y)\mapsto (xy + 1)^2 \\ K_-: (x,y)\mapsto (xy - 1)^2 \end{cases} $$
Both $K_+$ and $K_-$ are positive definite kernel and therefore both have an RKHS. Shouldn't $K_-$ be included in the polynomial kernel definition?
Another question I have about this example is: although it is quite easy to find $K_+$'s RKHS, it seems way more difficult to find a closed form expression of $K_-$'s RKHS.
An idea could be to see $K_-$ as a $90º$ rotation of $K_+$, but I couldn't get from this to a proper expression of the associated Hilbert space.
Any help would be greatly appreciated :)