Multigraded Koszul complexes, filter-regular sequences and lower bounds for the multiplicity of the resultant

Abstract

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.