Suppose I have a set S and I want to find a polynomial p such that p(s)=0modn if s∈S, and that it is non-zero modulo n otherwise.

In the literature such an S is sometimes called a root set (see http://www.sciencedirect.com/science/article/pii/S0195669896901249).

Finding such a polynomial is always possible if n is a prime number, but if p is a non-prime, it may not be.

For example, if S={1,2} and we try to find a polynomial p such that p(1)=p(2)=0mod6, then this implies that p(4)=p(5)=0mod6. So {1,2} is not a root set modulo 6.

Is there some characterization of root sets (modulo a certain integer)?

Do you maybe have useful references for me?

**Answer**

**Attribution***Source : Link , Question Author : Astrid Pieterse , Answer Author : Community*