The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ has $a_k \in M$ for $k < n$ and $a_n \in R^* = M^c$, then
$$
f = (z^n + b_{n-1}z^{n-1} + \cdots + b_0)u
$$
where $b_k \in M$ and $u$ is a unit in $R[[z]]$.
I need an explicit algorithm for calculating this Weierstrass polynomial (or distinguished polynomial) for a given $f$. In my case the coefficient ring is $R = \mathbb Z_3[[x]]$, formal power series over the 3-adics. So any algorithm would have to be robust enough to handle these coefficients.
Does anyone know of such an algorithm for a math software package?
Best Answer
If you're still interested in the answer to this...I also needed an explicit algorithm for calculating associated Weierstrass polynomials and provide two such algorithms in http://arxiv.org/abs/1107.4860v2, Algorithms 5.2 and 5.4. The first one is simplest and is based on Manin's method and a result by Sumida.