RuppertMatrix - Maple Help

PolynomialTools[Approximate]

 RuppertMatrix
 construct the Ruppert matrix of a polynomial

 Calling Sequence RuppertMatrix(F, vars) RuppertMatrix(F, vars, rtableoptions=[options])

Parameters

 F - polynom vars - set or list of variables options - (optional) options that are passed to the Matrix constructor

Description

 • The RuppertMatrix command returns a Matrix derived from the coefficients of F which counts the factors of F over the complex numbers with the dimension of its null space (its rank deficiency).
 • The Ruppert matrix is defined, for a bivariate polynomial $f$ over $\left[x,y\right]$, as the matrix of the linear system resulting from the partial differential equation $\frac{\partial }{\partial x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{h}_{y}}{f}\right)=\frac{\partial }{\partial y}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{g}{f}\right)$ where $g$ and ${h}_{y}$ are unknown polynomials of the same degree as $f$ in each variable except $\mathrm{degree}\left(g,x\right)\le \mathrm{degree}\left(f,x\right)-1$ and $\mathrm{degree}\left({h}_{y},y\right)\le \mathrm{degree}\left(f,y\right)-1$
 • The multivariate version of the Ruppert matrix is defined similarly, just adding an unknown polynomial ${h}_{{x}_{i}}$ and one equation for each additional variable ${x}_{i}$.
 • In this implementation the unknowns are additionally constrained so that they have total degree less than or equal to $f$. This results in a smaller matrix with the same property on its rank.
 • This routine is called by Factor.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right):$
 > $\mathrm{R1}≔\mathrm{RuppertMatrix}\left({x}^{2}+{y}^{2}-1,\left[x,y\right]\right)$
 ${\mathrm{R1}}{≔}\left[\begin{array}{cccccc}{0}& {0}& {-1}& {0}& {1}& {0}\\ {0}& {0}& {0}& {2}& {0}& {0}\\ {-2}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {1}& {0}\\ {0}& {-2}& {0}& {0}& {0}& {2}\\ {0}& {0}& {-1}& {0}& {-1}& {0}\end{array}\right]$ (1)

Polynomial is irreducible, so the rank deficiency is one

 > $\mathrm{min}\left(\mathrm{upperbound}\left(\mathrm{R1}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{R1}\right)$
 ${1}$ (2)

rtable options are passed to the matrix constructor; the datatype option is most useful for approximate factorization

 > $\mathrm{RuppertMatrix}\left({x}^{2}+{y}^{2}-1,\left[x,y\right],\mathrm{rtableoptions}=\left[\mathrm{datatype}=\mathrm{float}\left[8\right]\right]\right)$
 $\left[\begin{array}{cccccc}{0.}& {0.}& {-1.}& {0.}& {1.}& {0.}\\ {0.}& {0.}& {0.}& {2.}& {0.}& {0.}\\ {-2.}& {0.}& {0.}& {0.}& {0.}& {0.}\\ {0.}& {0.}& {1.}& {0.}& {1.}& {0.}\\ {0.}& {-2.}& {0.}& {0.}& {0.}& {2.}\\ {0.}& {0.}& {-1.}& {0.}& {-1.}& {0.}\end{array}\right]$ (3)

Many variables

 > $\mathrm{R2}≔\mathrm{RuppertMatrix}\left(\left({x}^{2}+{y}^{2}-1\right)\left({x}^{2}+{z}^{2}-1\right),\left[x,y,z\right]\right)$
 ${\mathrm{R2}}{≔}\begin{array}{c}\left[\begin{array}{ccccccccccc}{0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {\dots }\\ {2}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {-2}& {0}& {1}& {0}& {0}& {0}& {\dots }\\ {0}& {2}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {\dots }\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {2}& {\dots }\\ {0}& {0}& {0}& {0}& {-1}& {0}& {0}& {0}& {0}& {0}& {\dots }\\ {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {}\end{array}\right]\\ \hfill {\text{168 × 48 Matrix}}\end{array}$ (4)
 > $\mathrm{min}\left(\mathrm{upperbound}\left(\mathrm{R2}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{R2}\right)$
 ${2}$ (5)

References

 Ruppert, W.M. "Reducibility of polynomials f(x,y) modulo p." Journal of Number Theory Vol. 77(1), (1999): 62-70.
 Gao, S. "Factoring multivariate polynomials via partial differential equations." Mathematics of Computation Vol. 72(242), (2002): 801-822.
 Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials using singular value decomposition." Journal of Symbolic Computation Vol. 43(5), (2008): 359-376.

Compatibility

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