  A polynomial of the form $x^\alpha - p(x)$, where the degree of $p$ is less
than the total degree of $x^\alpha$, 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 $d=3$, extremal polynomial for $(x_1x_2x_3)^k$ on
the ball and the sphere is found for $k=2$ and 4. For $d \ge 3$, a family of
polynomials of the form $(x_1... x_d)^2 - p(x)$ is explicit given and proved to
be the least deviation from zero for $d =3,4,5$, and it is conjectured to be
the least deviation for all $d$.
