Counting Roots of Polynomials over Z/p^2Z
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Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime p ∈Z and f ∈ ( Z/p^n Z ) [x] any nonzero polynomial of degree d whose coefficients are not all divisible by p. For the case n=2, we prove a new efficient algorithm to count the roots of f in Z/p^2Z within time polynomial in (d+size(f)+p), and record a concise formula for the number of roots, formulated by Cheng, Gao, Rojas, and Wan.
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