  We study L^p-L^r restriction estimates for algebraic varieties in
d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if
the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2
Stein-Tomas restriction result can be improved to the L^{(2d+4)/(d+4)}-L^2
estimate for both spheres and paraboloids in finite fields. In this paper we
show that the conjectured L^p-L^2 restriction estimate holds in the specific
case when test functions under consideration are restricted to d-coordinate
functions or homogeneous functions of degree zero. To deduce our result, we use
the connection between the restriction phenomena for our varieties in $d$
dimensions and those for homogeneous varieties in (d+1)dimensions.
