  As an extension of previous ungraded work, we define a graded $p$-polar ring
to be an analog of a graded commutative ring where multiplication is only
allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show
that the free affine $p$-adic group scheme functor, as well as the free formal
group functor, defined on $k$-algebras for a perfect field $k$ of
characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the
same is true for any affine $p$-adic or formal group functor, in particular for
the functor of $p$-typical Witt vectors. As an application, we show that the
latter is free on the $p$-polar affine line.
