Zeros of polynomials modulo a non-prime

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

In the literature such an S is sometimes called a root set (see

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?


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

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