LaTeX4Web 1.4 OUTPUT
Similarly to Newton¢s Method, in p-adic analysis we find Hensel¢s Lemma, which allows to find the p-adic expansion of any given polynomial. Let f(x) be a polynomial function and we search its roots in the p-adic ring. First we need to define p. Then we find x_{1} such that f(x_{1}) º 0 mod p. Then we look for x_{2}. We know that it is of the form x_{2} = x_{1} + pt and we need to determine t. We want that x_{2} º x_{1} modulo p. By the Taylor expansion we know that f(x_{2}) = f(x_{1} + pt) = f(x_{1}) + ptf¢(x_{1}) +... We do not need to write all the expansion because f(x_{1}) + ptf¢(x_{1}) mod p^{2}
. Since we know that f(x_{1}) º 0 mod p we can divide the previous expression by p: f(x_{1})/p + tf¢(x_{1}) º 0 mod p. We can find t from this expression and then we can find x_{2} since it was defined as x_{1} + pt. This steps can be done recursively similar to Newton¢s Method. In general, we use these two expressions: f(x_{n-1})/p^{n-1}
+ tf¢(x_{n-1}) º 0 mod p x_{n} = x_{n-1} + tp^{n-1}
mod p^{n}
. Also similar to Newton¢s Method, the necessary condition for Hensel¢s Lemma to work is that f¢(x) ¹ 0 mod p. Also, the coefficients of the polynomials need to be reduced modulo m. We can take as an example the equation x^{2}
= 5. We know that x_{1} equals 4 and then x_{2} = x_{1} + pt = 4 + 11t mod p^{2}
. Then we find t by using the previous recursive expressions 11/11 + 8t º 0 mod p. We see that t equals 4. Then we know that x_{2} = 4 + 11t = 4 + 11 · 4 = 48. Using this we obtain the different a_{i} which are 4, 4, 10, 4, 0...
We also provide the code for Hensel's Lemma. The program will return "ERROR" if the derivative becomes zero. In the code it is possible to change until what prime should the program find expansions.