When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that the sum of $g_i\cdot f_i$ is $1$, (Nullstellensatz). However, this theorem doesn't seem to have any practical value, consider the following problem:

Is there any nontrivial criterion such that we can determine whether some polynomials (chosen randomly) have common zeros or not by using it? Of course, I don't want to use the Grobner base theory, what I need is just a criterion involving information on the algebraic properties of the given polynomials, something just like Eisensten's criterion on irreducibility.

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