changing parent rings of polynomials
I would like to be able to change the rings sage polynomials live in, so that the number of variables in the polynomial matches the number of generators of the ring. For example, if I have the polynomial x^4 in the Multivariate Polynomial Ring in x, y, z over Rational Field, I would like to get it (or a copy of it) into the Multivariate Polynomial Ring in x over Rational Field. Is there a simple way to do this?
On a similar note, when going in the opposite direction, sage recognizes R.<x> = QQ as a different type of object than R.<x,y> = QQ, which makes my life complicated. Is there a way to convert a polynomial in a single variable ring ( like x^4 in single variable QQ[x]) into the same polynomial but with a multivariable parent (like x^4 in a Multivariate Polynomial Ring QQ[x])?