 SylvesterMatrix - Maple Help

PolynomialTools[Approximate]

 SylvesterMatrix
 construct a Sylvester matrix of a polynomial Calling Sequence SylvesterMatrix(F, G, vars) SylvesterMatrix(F, G, vars, degree) SylvesterMatrix(F, G, vars, degree, rtableoptions=[options]) Parameters

 F - polynom G - polynom vars - set or list of variables degree - (optional) non-negative integer, defaults to 1 options - (optional) options that are passed to the Matrix constructor Description

 • The SylvesterMatrix command is a generalized and multivariate version of the LinearAlgebra:-SylvesterMatrix command.
 • A (generalized) Sylvester matrix is 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. Examples

 > $\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right):$
 > $f≔{x}^{2}+{y}^{2}-1;$$g≔{x}^{2}+xy+y+1$
 ${f}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}$
 ${g}{≔}{{x}}^{{2}}{+}{x}{}{y}{+}{y}{+}{1}$ (1)
 > $\mathrm{S1}≔\mathrm{SylvesterMatrix}\left(f,g,\left[x,y\right]\right)$
 ${\mathrm{S1}}{≔}\left[\begin{array}{cccccc}{-1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {-1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {-1}& {1}& {0}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {0}\\ {1}& {0}& {0}& {0}& {0}& {1}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {1}& {1}\\ {0}& {1}& {0}& {0}& {0}& {1}\\ {0}& {0}& {1}& {0}& {0}& {0}\end{array}\right]$ (2)

Maximal rank, means degree( gcd(f,g) ) < 1

 > $\mathrm{min}\left(\mathrm{upperbound}\left(\mathrm{S1}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S1}\right)$
 ${0}$ (3)
 > $\mathrm{SylvesterMatrix}\left(f,g,\left[x,y\right],2,'\mathrm{rtableoptions}'=\left[\mathrm{datatype}=\mathrm{complex}\left[8\right]\right]\right)$
 $\left[\begin{array}{cc}{-1.}{+}{0.}{}{I}& {1.}{+}{0.}{}{I}\\ {0.}{+}{0.}{}{I}& {0.}{+}{0.}{}{I}\\ {0.}{+}{0.}{}{I}& {1.}{+}{0.}{}{I}\\ {1.}{+}{0.}{}{I}& {1.}{+}{0.}{}{I}\\ {0.}{+}{0.}{}{I}& {1.}{+}{0.}{}{I}\\ {1.}{+}{0.}{}{I}& {0.}{+}{0.}{}{I}\end{array}\right]$ (4)
 > $d≔{x}^{2}-{y}^{2}+1;$$\mathrm{f1}≔fd;$$\mathrm{g1}≔gd$
 ${d}{≔}{{x}}^{{2}}{-}{{y}}^{{2}}{+}{1}$
 ${\mathrm{f1}}{≔}\left({{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}\right){}\left({{x}}^{{2}}{-}{{y}}^{{2}}{+}{1}\right)$
 ${\mathrm{g1}}{≔}\left({{x}}^{{2}}{+}{x}{}{y}{+}{y}{+}{1}\right){}\left({{x}}^{{2}}{-}{{y}}^{{2}}{+}{1}\right)$ (5)

Maximal rank, means degree( gcd(f,g) ) < 3

 > $\mathrm{S2}≔\mathrm{SylvesterMatrix}\left(\mathrm{f1},\mathrm{g1},\left[x,y\right],3\right)$
 ${\mathrm{S2}}{≔}\begin{array}{c}\left[\begin{array}{cccccc}{-1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {-1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {-1}& {1}& {0}& {1}\\ {0}& {0}& {0}& {2}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {0}\\ {2}& {0}& {0}& {-1}& {0}& {1}\\ {0}& {0}& {0}& {0}& {2}& {0}\\ {0}& {0}& {0}& {1}& {1}& {2}\\ {0}& {2}& {0}& {0}& {-1}& {1}\\ {0}& {0}& {2}& {-1}& {0}& {-1}\\ {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}\end{array}\right]\\ \hfill {\text{21 × 6 Matrix}}\end{array}$ (6)

Maximal rank, means degree( gcd(f,g) ) < 3

 > $\mathrm{min}\left(\mathrm{upperbound}\left(\mathrm{S2}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S2}\right)$
 ${0}$ (7)
 > $\mathrm{S3}≔\mathrm{SylvesterMatrix}\left(\mathrm{f1},\mathrm{g1},\left[x,y\right],2\right)$
 ${\mathrm{S3}}{≔}\begin{array}{c}\left[\begin{array}{ccccccccccc}{-1}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {\dots }\\ {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {\dots }\\ {0}& {0}& {-1}& {0}& {0}& {0}& {1}& {0}& {1}& {0}& {\dots }\\ {0}& {0}& {0}& {-1}& {0}& {0}& {2}& {0}& {0}& {1}& {\dots }\\ {0}& {0}& {0}& {0}& {-1}& {0}& {1}& {1}& {0}& {0}& {\dots }\\ {2}& {0}& {0}& {0}& {0}& {-1}& {-1}& {0}& {1}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {1}& {2}& {1}& {\dots }\\ {0}& {2}& {0}& {0}& {0}& {0}& {0}& {-1}& {1}& {0}& {\dots }\\ {0}& {0}& {2}& {0}& {0}& {0}& {-1}& {0}& {-1}& {0}& {\dots }\\ {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {}\end{array}\right]\\ \hfill {\text{28 × 12 Matrix}}\end{array}$ (8)

Rank deficiency of exactly 1, means degree( gcd(f,g) ) = 2 exactly

 > $\mathrm{min}\left(\mathrm{upperbound}\left(\mathrm{S3}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{S3}\right)$
 ${1}$ (9) Compatibility

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