  We obtain $L^2$ decay estimates in $\lambda$ for oscillatory integral
operators whose phase functions are homogeneous polynomials of degree m and
satisfy various genericity assumptions. The decay rates obtained are optimal in
the case of (2+2)--dimensions for any m while, in higher dimensions, the result
is sharp for m sufficiently large. The proof for large $m$ follows from
essentially algebraic considerations. For cubics in (2+2)--dimensions, the
proof involves decomposing the operator near the conic zero variety of the
determinant of the Hessian of the phase function, using an elaboration of the
general approach of Phong and Stein [1994].
