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Root Sets of Polynomials Modulo Prime Powers

✍ Scribed by Davesh Maulik

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
194 KB
Volume
93
Category
Article
ISSN
0097-3165

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✦ Synopsis

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 p k for fixed k is a polynomial in p, raised to the p th power. Moreover, we show that the leading term of this polynomial is

!) &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.

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