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 $\frac{\partial }{\partial x}\left(\frac{h_y}{f}\right)=\frac{\partial }{\partial y}\left(\frac{g}{f}\right)$ 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.

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

$\mathrm{R1}≔\mathrm{RuppertMatrix}\left(x^2+y^2-1\,\left[x\,y\right]\right)$

$\mathrm{R1}≔\left[\begin{array}{cccccc}0& 0& \mathrm{-1}& 0& 1& 0\\ 0& 0& 0& 2& 0& 0\\ \mathrm{-2}& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& \mathrm{-2}& 0& 0& 0& 2\\ 0& 0& \mathrm{-1}& 0& \mathrm{-1}& 0\end{array}\right]$

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

$1$

$\mathrm{RuppertMatrix}\left(x^2+y^2-1\,\left[x\,y\right]\,\mathrm{rtableoptions}=\left[\mathrm{datatype}=\mathrm{float}\left[8\right]\right]\right)$

$\left[\begin{array}{cccccc}0.& 0.& \mathrm{-1.}& 0.& 1.& 0.\\ 0.& 0.& 0.& 2.& 0.& 0.\\ \mathrm{-2.}& 0.& 0.& 0.& 0.& 0.\\ 0.& 0.& 1.& 0.& 1.& 0.\\ 0.& \mathrm{-2.}& 0.& 0.& 0.& 2.\\ 0.& 0.& \mathrm{-1.}& 0.& \mathrm{-1.}& 0.\end{array}\right]$

$\mathrm{R2}≔\mathrm{RuppertMatrix}\left(\left(x^2+y^2-1\right)\left(x^2+z^2-1\right)\,\left[x\,y\,z\right]\right)$

$\mathrm{R2}≔\begin{array}{c}\left[\begin{array}{ccccccccccc}0& 0& 0& 0& 1& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& \dots \\ 2& 0& 0& 0& 0& 0& 0& 2& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& \mathrm{-2}& 0& 1& 0& 0& 0& \dots \\ 0& 2& 0& 0& 0& 0& 0& 0& 2& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& \dots \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& \dots \\ 0& 0& 0& 0& \mathrm{-1}& 0& 0& 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{168 × 48 Matrix}\end{array}$

$\min \left(\mathrm{upperbound}\left(\mathrm{R2}\right)\right)-\mathrm{LinearAlgebra}:-\mathrm{Rank}\left(\mathrm{R2}\right)$

$2$

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.

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

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