  Let $f$ be a homogeneous polynomial with rational coefficients in $d$
variables. We prove several results concerning uniform simultaneous
approximation to points on the graph of $f$, as well as on the hypersurface
$\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case
$f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.
