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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.

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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.

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