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Let's examine a simple case:

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Dg :=  [p3 - 3*p2 + 3*p1 - p0,3*p2 - 6*p1 + 3*p0,3*p1 - 3*p0,p0]

\label{eq1}\left[{p 3 -{3 \  p 2}+{3 \  p 1}- p 0}, \:{{3 \  p 2}-{6 \  p 1}+{3 \  p 0}}, \:{{3 \  p 1}-{3 \  p 0}}, \: p 0 \right](1)
Type: List(Polynomial(Integer))

Now calculate coefficients in two ways:

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map(coefficients, Dg::List MPOLY([p0,p1,p2,p3], INT))

\label{eq2}\left[{\left[ - 1, \: 3, \: - 3, \: 1 \right]}, \:{\left[ 3, \: - 6, \: 3 \right]}, \:{\left[ - 3, \: 3 \right]}, \:{\left[ 1 \right]}\right](2)
Type: List(List(Integer))

and

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map(coefficients, Dg::List MPOLY([p3,p2,p1,p0], INT))

\label{eq3}\left[{\left[ 1, \: - 3, \: 3, \: - 1 \right]}, \:{\left[ 3, \: - 6, \: 3 \right]}, \:{\left[ 3, \: - 3 \right]}, \:{\left[ 1 \right]}\right](3)
Type: List(List(Integer))

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 FriCAS book function coefficients does not include zeros in the list. Furthermore it does not say explicitly in what order the coefficients themselves are listed. In fact coefficients are listed in monomial order. If one need to both coeffcients and variables one can use monomials to get them together, or use degree on monomials to extract variables.

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pol1 := first(Dg)

\label{eq4}p 3 -{3 \  p 2}+{3 \  p 1}- p 0(4)
Type: Polynomial(Integer)
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monomials(pol1)

\label{eq5}\left[ p 3, \: -{3 \  p 2}, \:{3 \  p 1}, \: - p 0 \right](5)
Type: List(Polynomial(Integer))
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coefficients(pol1)

\label{eq6}\left[ 1, \: - 3, \: 3, \: - 1 \right](6)
Type: List(Integer)
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map(degree, monomials(pol1))

\label{eq7}\left[ p 3, \: p 2, \: p 1, \: p 0 \right](7)
Type: List(IndexedExponents?(Symbol))
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pol2 := pol1::MPOLY([p0,p1,p2,p3], INT)

\label{eq8}- p 0 +{3 \  p 1}-{3 \  p 2}+ p 3(8)
Type: MultivariatePolynomial?([p0,p1,p2,p3],Integer)
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monomials(pol2)

\label{eq9}\left[ - p 0, \:{3 \  p 1}, \: -{3 \  p 2}, \: p 3 \right](9)
Type: List(MultivariatePolynomial?([p0,p1,p2,p3],Integer))
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coefficients(pol2)

\label{eq10}\left[ - 1, \: 3, \: - 3, \: 1 \right](10)
Type: List(Integer)
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map(degree, monomials(pol2))

\label{eq11}\left[ p 0, \: p 1, \: p 2, \: p 3 \right](11)
Type: List(IndexedExponents?(OrderedVariableList([p0,p1,p2,p3])))

Remember that there are also terms of degree higher than 1. In fact the two multivariate polynomials that you used above are formally identical

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(Dg::List MPOLY([p0,p1,p2,p3], INT)=Dg::List MPOLY([p3,p2,p1,p0], INT))::Boolean

\label{eq12} \mbox{\rm true} (12)
Type: Boolean

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:

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[[coefficient(p,x,1) for x in [p0,p1,p2,p3]] for p in Dg]

\label{eq13}\left[{\left[ - 1, \: 3, \: - 3, \: 1 \right]}, \:{\left[ 3, \: - 6, \: 3, \: 0 \right]}, \:{\left[ - 3, \: 3, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}\right](13)
Type: List(List(Polynomial(Integer)))

And of course you can use this to produce a matrix.

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matrix %

\label{eq14}\left[ 
\begin{array}{cccc}
- 1 & 3 & - 3 & 1 
\
3 & - 6 & 3 & 0 
\
- 3 & 3 & 0 & 0 
\
1 & 0 & 0 & 0 
(14)
Type: Matrix(Polynomial(Integer))

Actually, unstated assumption in FriCAS is that multivariate polynomials are sparse, so omiting zeros is desirable. For univariate polynomials one can use vectorise. Choosing p0 as main variable and converting to univariate polynomial in one variable we get:

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[vectorise(univariate(p, p0), 2) for p in Dg]

\label{eq15}\left[{\left[{p 3 -{3 \  p 2}+{3 \  p 1}}, \: - 1 \right]}, \:{\left[{{3 \  p 2}-{6 \  p 1}}, \: 3 \right]}, \:{\left[{3 \  p 1}, \: - 3 \right]}, \:{\left[ 0, \: 1 \right]}\right](15)
Type: List(Vector(Polynomial(Integer)))




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