  We prove that there are no regular algebraic hypersurfaces with non-zero
constant mean curvature in the Euclidean space $\mathbb R^{n+1}$, $n\geq 2$,
defined by polynomials of odd degree. Also we prove that the hyperspheres and
the round cylinders are the only regular algebraic hypersurfaces with non-zero
constant mean curvature in $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials
of degree less than or equal to three. These results give partial answers to a
question raised by Barbosa and do Carmo.
