User r. nendorf - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:50:54Z http://mathoverflow.net/feeds/user/10135 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42564/algorithm-for-weierstrass-preparation-theorem-for-formal-power-series Algorithm for Weierstrass Preparation Theorem for Formal Power Series R. Nendorf 2010-10-18T01:05:54Z 2011-08-18T14:07:38Z <p>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 $k &lt; n$ and $a_n \in R^* = M^c$, then</p> <p>$$f = (z^n + b_{n-1}z^{n-1} + \cdots + b_0)u$$ where $b_k \in M$ and $u$ is a unit in $R[[z]]$.</p> <p>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.</p> <p>Does anyone know of such an algorithm for a math software package?</p> http://mathoverflow.net/questions/42564/algorithm-for-weierstrass-preparation-theorem-for-formal-power-series Comment by R. Nendorf R. Nendorf 2010-11-08T18:37:04Z 2010-11-08T18:37:04Z It seems that Manin's method suggests an algoriithm to approximate the unit power series (that I called &quot;u&quot; above) to any finite degree of precision. But being interested in the polynomial I would need to calculate u exactly so that I could find f*(u^-1). The package WEIER seems to only work for field coefficients, though I had trouble executing it. Entreaties to the FriCAS development group haven't produced a response.