The SylvesterMatrix command is a generalized and multivariate version of the LinearAlgebra:-SylvesterMatrix command.

A (generalized) Sylvester matrix is a matrix that has full rank only if the input polynomials have a greatest common divisor of total degree less than degree (1 by default).

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

The approximate polynomial division command GCD solves an approximate nullspace 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$

$f≔x^2+y^2-1$

$g≔x^2+xy+y+1$

$\mathrm{S1}≔\mathrm{SylvesterMatrix}\left(f\,g\,\left[x\,y\right]\right)$

$\mathrm{S1}≔\left[\begin{array}{cccccc}\mathrm{-1}& 0& 0& 1& 0& 0\\ 0& \mathrm{-1}& 0& 0& 1& 0\\ 0& 0& \mathrm{-1}& 1& 0& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 1& 0\\ 1& 0& 0& 0& 0& 1\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 1& 1\\ 0& 1& 0& 0& 0& 1\\ 0& 0& 1& 0& 0& 0\end{array}\right]$

$\min \left(\mathrm{upperbound}\left(\mathrm{S1}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S1}\right)$

$0$

$\mathrm{SylvesterMatrix}\left(f\,g\,\left[x\,y\right]\,2\,'\mathrm{rtableoptions}'=\left[\mathrm{datatype}=\mathrm{complex}\left[8\right]\right]\right)$

$\left[\begin{array}{cc}\mathrm{-1.}+0.\mathrm{I}& 1.+0.\mathrm{I}\\ 0.+0.\mathrm{I}& 0.+0.\mathrm{I}\\ 0.+0.\mathrm{I}& 1.+0.\mathrm{I}\\ 1.+0.\mathrm{I}& 1.+0.\mathrm{I}\\ 0.+0.\mathrm{I}& 1.+0.\mathrm{I}\\ 1.+0.\mathrm{I}& 0.+0.\mathrm{I}\end{array}\right]$

$d≔x^2-y^2+1\;$$\mathrm{f1}≔fd\;$$\mathrm{g1}≔gd$

$d≔x^2-y^2+1$

$\mathrm{f1}≔\left(x^2+y^2-1\right)\left(x^2-y^2+1\right)$

$\mathrm{g1}≔\left(x^2+xy+y+1\right)\left(x^2-y^2+1\right)$

$\mathrm{S2}≔\mathrm{SylvesterMatrix}\left(\mathrm{f1}\,\mathrm{g1}\,\left[x\,y\right]\,3\right)$

$\mathrm{S2}≔\begin{array}{c}\left[\begin{array}{cccccc}\mathrm{-1}& 0& 0& 1& 0& 0\\ 0& \mathrm{-1}& 0& 0& 1& 0\\ 0& 0& \mathrm{-1}& 1& 0& 1\\ 0& 0& 0& 2& 0& 0\\ 0& 0& 0& 1& 1& 0\\ 2& 0& 0& \mathrm{-1}& 0& 1\\ 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 2& 0& 0& \mathrm{-1}& 1\\ 0& 0& 2& \mathrm{-1}& 0& \mathrm{-1}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]\\ \hfill \text{21 × 6 Matrix}\end{array}$

$\min \left(\mathrm{upperbound}\left(\mathrm{S2}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S2}\right)$

$0$

$\mathrm{S3}≔\mathrm{SylvesterMatrix}\left(\mathrm{f1}\,\mathrm{g1}\,\left[x\,y\right]\,2\right)$

$\mathrm{S3}≔\begin{array}{c}\left[\begin{array}{ccccccccccc}\mathrm{-1}& 0& 0& 0& 0& 0& 1& 0& 0& 0& \dots \\ 0& \mathrm{-1}& 0& 0& 0& 0& 0& 1& 0& 0& \dots \\ 0& 0& \mathrm{-1}& 0& 0& 0& 1& 0& 1& 0& \dots \\ 0& 0& 0& \mathrm{-1}& 0& 0& 2& 0& 0& 1& \dots \\ 0& 0& 0& 0& \mathrm{-1}& 0& 1& 1& 0& 0& \dots \\ 2& 0& 0& 0& 0& \mathrm{-1}& \mathrm{-1}& 0& 1& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 2& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& 1& 1& 2& 1& \dots \\ 0& 2& 0& 0& 0& 0& 0& \mathrm{-1}& 1& 0& \dots \\ 0& 0& 2& 0& 0& 0& \mathrm{-1}& 0& \mathrm{-1}& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{28 × 12 Matrix}\end{array}$

$\min \left(\mathrm{upperbound}\left(\mathrm{S3}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S3}\right)$

$1$

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

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