Find the polynomial root modulo 2 ^ r

Question

I have a polynomial P and I would like to find y that P (y) = 0 modulo 2 ^ r.

I tried something along the Hensel lift line, but I don’t know if this can even work, due to the usual condition f '(y mod 2)! = 0 mod 2, which is usually not true.

Is there any other algorithm? Or can the Hensel lift work be changed?

Thanks in advance

Answer 1

Suppose you have a solution asuch that f(a) = 0 mod 2^p. To make a Hensel lift to get a solution mod 2^(p+1), you need to decide

f'(a)*t = -f(a)/2^(p+1) mod 2

for t.

If f'(a) = 0 mod 2, two possibilities are possible:

2 f(a)/2^(p+1), mod 2^(p+1) ( 2) a.

2 f(a)/2^(p+1), 0, 1 t, , mod 2^r.

, a [0,2^p) , , t, f(x) f'(x) x=a, x=a mod 2.