  We present bounds for the degree and the height of the polynomials arising in
some central problems in effective algebraic geometry including the
implicitation of rational maps and the effective Nullstellensatz over a
variety. Our treatment is based on arithmetic intersection theory in products
of projective spaces and extends to the arithmetic setting constructions and
results due to Jelonek. A key role is played by the notion of canonical mixed
height of a multiprojective variety. We study this notion from the point of
view of resultant theory and establish some of its basic properties, including
its behavior with respect to intersections, projections and products. We obtain
analogous results for the function field case, including a parametric
Nullstellensatz.
