# Resultant in many variables

Suppose $$P_1,\ldots,P_n$$ are homogeneous polynomials in $$\mathbb C[x_0,\ldots,x_n]$$ of degrees $$d_1,\ldots,d_n\ge 1$$. These define hypersurfaces $$H_1,\ldots,H_n\subset\mathbb P^n$$. Is there a nonzero polynomial in the coefficients of $$P_1,\ldots,P_n$$ which vanishes whenever when the common intersection $$H_1\cap\cdots\cap H_n$$ is not zero dimensional? If so, what is an explicit method to determine this polynomial? Any references dealing with this would also be appreciated.

Let $$F_i$$ be a homogeneous polynomial of degree $$d_i$$ defining $$H_i$$. Let $$U(x)$$ be a linear form. Then consider the multivariate resultant $${\rm Res}(F_1,\ldots,F_n,U)$$ which, for fixed $$F_i$$'s is a homogeneous polynomial of degree $$d_1\cdots d_n$$ in the coefficients of $$U$$. You condition is equivalent to the vanishing of this polynomial identically in $$U$$. For information about this multivariate resultant see, e.g., "Explicit formulas for the multivariate resultant" by D'Andrea and Dickenstein, and references therein. Computing them explicitly is very hard in general.