# Finding a characteristic for which the zero-locus of an ideal is not empty

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\overline{\mathbb{F}_p}[x_1, \dots, x_n]$ for some prime $p$. One way to do this is to calculate the Gröbner basis of the ideal $$I = (f_1, \dots, f_m) \subseteq \mathbb{Z}[x_1, \dots, x_n].$$ If this basis contains some integer $k > 1$, then we can deduce that the ideal is not trivial inside $\overline{\mathbb{F}_p}[x_1, \dots, x_n]$ for any prime $p$ dividing $k$. If this basis contains a $1$, we find that $I$ is trivial in any characteristic, while if the basis does not contain an integer, we find that $I$ is non-trivial characteristic $0$. Since the computation of a Gröbner basis can be very difficult, I was wondering if there is another more direct way of computing this integer $k$ (whether or not it is $0$,$1$ or $>1$). Maybe there is another way altogether to determine whether or not a set of equations has a root in some characteristic.

I am also very interested in software that is capable of doing such a computation. So far I've used Mathematica and Sage for Gröbner basis computations, but I am not sure if these packages are the most well suited for the job.