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.



Resultant in many variables
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"](https://arxiv.org/abs/math/0007036) by D'Andrea and Dickenstein, and references therein.
Computing them explicitly is very hard in general.


