I was trying to solve this problem:
In a previous exercise I had proven that if two polynomials $f,g\in A[x,y]$ are coprime then the algebraic variety $V(f,g)$ is finite. Using this, what I did in my first attempt to solve the problem was to say: since the gcd of $f_1,...,f_m$ is a constant, some $f_i$ and $f_j$ must be coprime and then $V(f_1,...,f_m)\subset V(f_i,f_j)$ is finite. However I have realised that this is wrong because it may happen that there is no common factor for $f_1,...,f_m$, but still there is a common factor for each pair $f_i,f_j$.
What would be the correct way to approach this problem? Thanks for your help.
Note: I am not sure if this is important, but we may assume $A$ is a field.