Root Sets of Polynomials Modulo Prime Powers

A subset R of the integers modulo n is defined to be a root set if it is the set of roots of some polynomial. Using the Chinese Remainder Theorem, the question of finding and counting root sets mod n is reduced to finding root sets modulo a prime power. In this paper, we provide a recursive construction for root sets modulo a prime power. We use this recursion to show that the number of root sets modulo pk for fixed k is a polynomial in p, raised to the pth power. Moreover, we show that the leading term of this polynomial is ckp⌊k2/4⌋ where ck=(k2!)−1 if k is even and ck=(k−12!)−1+(k+12!)−1 if k is odd, thus giving an asymptotic estimate on the number of root sets for fixed k. Finally, we generalize these results to arbitrary Dedekind domains.