# Difference between polynomial regression and polynomial kernel

A few answers on SO suggested that a polynomial transformation and a regularized regression can be used instead of a polynomial kernel regression. What's the difference between them?

I thought polynomial feature transformation of degree d has combinatorially many parameters, while polynomial kernel has no parameters at all once d is fixed. So without regularization, polynomial feature transformation should result in a much higher complexity / flexibility of the model. Does L1 or L2 regularization after adding polynomial features make two approaches somewhat similar?