Excellent exactly what I was looking for.

@Max can't you plug them into each other? The value of x can be plugged into the values for x^i? That should give $n-1$ conditions.

There is no formula for a GCD though no? I was looking for a situation where you can write down an ideal in the coefficients of the 3 polynomials and the ideal vanishes precisely on those points where those 3 polynomials share a common root. @Derek your example seems mysterious to me even for two polynomials. But we do know the resultant works there.

Thanks for the answer! I also wanted to understand how the abelian group of units behaves? Is there any reference you would recommend?

Yeah, I am looking for flat and injective examples. In particular, I am thinking of cases of the form $\mathbb{Z}[x]/(p^k,f(x))$ where you can take $f$ to be of degree $n$ and irreducible degree $n$ mod $p$.

Ahh didn't know the problem was still open. Thanks for the replies.