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## [Axiom-developer] [Polynomial Coefficients] (new)

 From: Bill Page Subject: [Axiom-developer] [Polynomial Coefficients] (new) Date: Tue, 21 Jun 2005 22:46:27 -0500

Changes
http://page.axiom-developer.org/zope/mathaction/PolynomialCoefficients/diff
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Let's examine a simple case:
\begin{axiom}
Dg :=  [p3 - 3*p2 + 3*p1 - p0,3*p2 - 6*p1 + 3*p0,3*p1 - 3*p0,p0]
\end{axiom}
Now calculate coefficients in two ways:
\begin{axiom}
map(coefficients, Dg::List MPOLY([p0,p1,p2,p3],?))
\end{axiom}
and
\begin{axiom}
map(coefficients, Dg::List MPOLY([p3,p2,p1,p0],?))
\end{axiom}
As you see the list are all reversed, but... they were not padded
with zeros. While in my opinion they should be - we have a given
order of variables, and alone 1 in the last list suggests that this
is a coefficient of p3 while it isn't. It is a coefficient of p0.

According to the Axiom book function 'coefficients' does not
include zeros in the list. Furthermore it does not say explicitly
in what order the coefficients themselves are listed. Remember
that there are also terms of degree higher than 1. In fact the
two multivariate polynomials that you used above are formally
identical
\begin{axiom}
(Dg::List MPOLY([p0,p1,p2,p3],?)=Dg::List MPOLY([p3,p2,p1,p0],?))::Boolean
\end{axiom}

To produce a list of coefficients of the terms of degree 1,
including the zeros and in a specific order, use the function
'coefficient' like this:
\begin{axiom}
[[coefficient(p,x,1) for x in [p0,p1,p2,p3]] for p in Dg]
\end{axiom}

And of course you can use this to produce a matrix.
\begin{axiom}
matrix %
\end{axiom}

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