The MahlerSystem(A, x, vn, vo) command computes the Mahler system of an m x n rectangular Matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q (that is, univariate polynomials in x with coefficients in Q(a1,...,an)), its residual R, and its closest normal point v.

The MahlerSystem(A, x, vn, vo, true) command returns the Mahler system, residual, closest normal point, the order of the Mahler system computed, and a list of indices indicating the nonzero columns of R.

If M = MahlerSystem(A, x, vn, vo) with the entries of A from $F_x$, the columns of M form a  module basis for the  (mathematical) module

in the sense that a module basis consists of $M\[*,i\],\mathrm{...},x^{v_i-1}M\[*,i\]$ for $i=1,...,n$ where n is the number of columns of M and v is the closest normal point to vn.

If the residual R is returned, it satisfies $A·M=x^{\mathrm{vo}}·R$, where $x^{\mathrm{vo}}$ is the diagonal matrix containing $x^{{\mathrm{vo}}_i}$ in entry $i,i$.

$\mathrm{with}\left(\mathrm{MatrixPolynomialAlgebra}\right)\:$

$A≔⟨⟨z^5-z^2-1\,z^3-2z^2+2z-2\,z+1⟩|⟨z^3-2z^2-1\,z^3-3z^2+3z-4\,2-z^3⟩⟩$

$A≔\left[\begin{array}{cc}{z}^5-z^2-1& z^3-2z^2-1\\ z^3-2z^2+2z-2& z^3-3z^2+3z-4\\ z+1& -z^3+2\end{array}\right]$

$\mathrm{vorder}≔\left[3\,5\,4\right]\:$

$M≔\mathrm{MahlerSystem}\left(A\,z\,\left[1\,3\right]\,\mathrm{vorder}\right)$

$M≔\left[\begin{array}{cc}-128z^3& 0\\ -16z^4+64z^3& -128z^5\end{array}\right]$

$\mathrm{map}\left(\mathrm{expand}\,A·M\right)$

$\left[\begin{array}{cc}-128z^8-16z^7+96z^6+16z^4+64z^3& -128z^8+256z^7+128z^5\\ -16z^7-16z^6+16z^5& -128z^8+384z^7-384z^6+512z^5\\ 16z^7-64z^6-160z^4& 128z^8-256z^5\end{array}\right]$

$M,R,v,\mathrm{vorder},\mathrm{nonzero}≔\mathrm{MahlerSystem}\left(A\,z\,\left[1\,3\right]\,\mathrm{vorder}\,\mathrm{true}\right)$

$M,R,v,\mathrm{vorder},\mathrm{nonzero}≔\left[\begin{array}{cc}-128z^3& 0\\ -16z^4+64z^3& -128z^5\end{array}\right],\left[\begin{array}{cc}-128z^5-16z^4+96z^3+16z+64& -128z^5+256z^4+128z^2\\ -16z^2-16z+16& -128z^3+384z^2-384z+512\\ 16z^3-64z^2-160& 128z^4-256z\end{array}\right],\left[\begin{array}{cc}3& 5\end{array}\right],\left[\begin{array}{ccc}3& 5& 4\end{array}\right],\left[1\,2\right]$

$W≔\mathrm{Matrix}\left(3\,3\,\left(i\,j\right)\mapsto \mathbf{if}i=j\mathbf{then}{z}^{\mathrm{vorder}\left[i\right]}\mathbf{else}0\mathbf{end}\mathbf{if}\right)\:$

$\mathrm{map}\left(\mathrm{expand}\,A·M-W·R\right)$

$\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

Beckermann, B. and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications. Vol. 22 No. 1, (2000): 114-144.