A polynomial of the form {\small $x^\alpha - p(x)$}, where the degree of
p is less than the total degree of {\small $x^\alpha$}, is said to
be least deviation from zero if it has the smallest uniform norm among all such
polynomials. We study polynomials of least deviation from zero over the unit
ball, the unit sphere, and the standard simplex. For {\small $d=3$}, extremal
polynomial for {\small $(x_1x_2x_3)^k$} on the ball and the sphere is found for
{\small $k=2$} and 4. For {\small $d \ge 3$}, a family of
polynomials of the form {\small $(x_1\cdots x_d)^2 - p(x)$} is explicitly given
and proved to be the least deviation from zero for {\small $d =3,4,5$}, and it
is conjectured to be the least deviation for all d.