  The R\'emond resultant attached to a multiprojective variety and a sequence
of multihomogeneous polynomials is a polynomial form in the coefficients of the
polynomials, which vanishes if and only if the polynomials have a common zero
on the variety. We demonstrate that this resultant can be computed as a Cayley
determinant of a multigraded Koszul complex, proving a key stabilization
property with the aid of local Hilbert functions and the notion of
filter-regular sequences. Then we prove that the R\'emond resultant vanishes,
under suitable hypotheses, with order at least equal to the number of common
zeros of the polynomials. More generally, we estimate the multiplicity of
resultants of multihomogeneous polynomials along prime ideals of the
coefficient ring, thus considering for example the order of $p$-adic vanishing.
Finally, we exhibit a corollary of this multiplicity estimate in the context of
interpolation on commutative algebraic groups, with applications to
Transcendental Number Theory.
