The ConvolutionMatrix command returns a Matrix that represents polynomial multiplication of F by a polynomial of the given degree (total degree) when the polynomial to multiply is represented as a vector as returned by CoefficientVector.

A Sylvester matrix can be considered to be a block matrix composed of two convolution matrices and the SylvesterMatrix command calls this one.

The approximate polynomial division command Divide solves up a least squares problem on the output of this command.

$\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right)\:$

$f≔x^2+y^2-1\;$$g≔x^2+xy+y+1$

$f≔x^2+y^2-1$

$g≔x^2+xy+y+1$

$\mathrm{fg}≔\mathrm{sort}\left(\mathrm{expand}\left(fg\right)\,\left[x\,y\right]\right)$

$\mathrm{fg}≔x^4+x^{3}y+x^2{y}^2+x{y}^3+x^{2}y+y^3-xy+y^2-y-1$

$\mathrm{Cf}≔\mathrm{ConvolutionMatrix}\left(f\,\left[x\,y\right]\,\mathrm{degree}\left(g\,\left\{x\,y\right\}\right)\right)$

$\mathrm{Cf}≔\begin{array}{c}\left[\begin{array}{cccccc}\mathrm{-1}& 0& 0& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0& 0& 0\\ 0& 0& \mathrm{-1}& 0& 0& 0\\ 1& 0& 0& \mathrm{-1}& 0& 0\\ 0& 0& 0& 0& \mathrm{-1}& 0\\ 1& 0& 0& 0& 0& \mathrm{-1}\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]\\ \hfill \text{15 × 6 Matrix}\end{array}$

$\mathrm{vg}≔\mathrm{PolynomialTools}:-\mathrm{CoefficientVector}\left(g\,\left[x\,y\right]\right)$

$\mathrm{vg}≔\left[\begin{array}{c}1\\ 0\\ 1\\ 1\\ 1\\ 0\end{array}\right]$

$\mathrm{vfg}≔\mathrm{Cf}·\mathrm{vg}$

$\mathrm{vfg}≔\begin{array}{c}\left[\begin{array}{c}\mathrm{-1}\\ 0\\ \mathrm{-1}\\ 0\\ \mathrm{-1}\\ 1\\ 0\\ 1\\ 0\\ 1\\ ⋮\end{array}\right]\\ \hfill \text{15 element Vector[column]}\end{array}$

$\mathrm{PolynomialTools}:-\mathrm{FromCoefficientVector}\left(\mathrm{vfg}\,\left[x\,y\right]\right)$

$x^4+x^{3}y+x^2{y}^2+x{y}^3+x^{2}y+y^3-xy+y^2-y-1$

The PolynomialTools:-Approximate:-ConvolutionMatrix command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.