# Why do we need the gamma parameter in the polynomial kernel of SVMs?

The polynomial kernel is sometimes defined as just: $$K(x,y):=(\left+c)^d$$ with two parameters: the degree $$d$$ and constant coefficient $$c$$.

But others (e.g., libsvm, and sklearn which uses libsvm) include a scaling parameter $$\gamma$$: $$K^\prime(x,y):=(\gamma \left+r)^d$$

Now it's fairly easy to see that this is redundant. Choose $$r=\gamma c$$, and we get $$K^\prime(x,y):=(\gamma \left+\gamma c)^d=\gamma^d(\left+c)^d=\gamma^d K(x,y)$$

But a constant scaling factor on the kernel does not matter for SVMs, does it?

I can imagine the following arguments: (1) consistency with RBF, where $$\gamma$$ is essential to scale the Gaussians (but I doubt you can choose the same value for both); (2) to avoid certain numerical range problems (c.f., libsvm guide mentions that the polynomial kernel can be problematic, but so can RBF) (3) other algorithms that do not learn weights - but which other algorithms work well with kernels and do not have weights to optimize?

Is there any theoretical reason why we should include $$\gamma$$ when teaching kernels?