  Polar codes that approach capacity at a near-optimal speed, namely with
scaling exponents close to $2$, have been shown possible for $q$-ary erasure
channels (Pfister and Urbanke), the BEC (Fazeli, Hassani, Mondelli, and Vardy),
all BMS channels (Guruswami, Riazanov, and Ye), and all DMCs (Wang and
Duursma). There is, nevertheless, a subtlety separating the last two papers
from the first two, namely the usage of multiple dynamic kernels in the
polarization process, which leads to increased complexity and fewer
opportunities to hardware-accelerate. This paper clarifies this subtlety,
providing a trade-off between the number of kernels in the construction and the
scaling exponent. We show that the number of kernels can be bounded by
$O(\ell^{3/\mu-1})$ where $\mu$ is the targeted scaling exponent and $\ell$ is
the kernel size. In particular, if one settles for scaling exponent approaching
$3$, a single kernel suffices, and to approach the optimal scaling exponent of
$2$, about $O(\sqrt{\ell})$ kernels suffice.
