The command ChangeOfCoordinates returns a list dec2 of regular chains of R2 which forms a Kalkbrener decomposition of the same saturated ideal as rc, after applying to this ideal the linear change of coordinates defined by M and V.

This linear change of coordinates maps the coordinates (given by the variables of the polynomial ring R) to the ordered variables given by V via the invertible linear transformation given by M.

In cases where the change of coordinates is a permutation, it can also be performed by the ChangeOfOrder command.

This command is part of the ChainTools package, so it can be used in the form ChangeOfCoordinates(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][ChangeOfCoordinates](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$F≔\left[5y^4-3\,20x-y+z\,x^5-y^5+3y+1\right]$

$F≔\left[5y^4-3\,20x-y+z\,x^5-y^5+3y+1\right]$

$\mathrm{vars}≔\left[z\,x\,y\right]$

$\mathrm{vars}≔\left[z\,x\,y\right]$

$\mathrm{vars2}≔\left[x\,y\,z\right]$

$\mathrm{vars2}≔\left[x\,y\,z\right]$

$M≔\mathrm{Matrix}\left(\left[\left[0\,1\,0\right]\,\left[0\,0\,1\right]\,\left[1\,0\,0\right]\right]\right)$

$M≔\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$

$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{dec}≔\mathrm{Triangularize}\left(F\,R\,'\mathrm{normalized}'='\mathrm{yes}'\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc}≔\mathrm{dec}\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}z+20x-y=0& \\ 5x^5+12y+5=0& \\ 5y^4-3=0& \end{array}\right.$

$\mathrm{dec2}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc}\,R\,M\,\mathrm{vars2}\right)$

$\mathrm{dec2}≔\left[\mathrm{regular\_chain}\,\mathrm{polynomial\_ring}\right]$

$\mathrm{rc2}≔\mathrm{dec2}\left[1\right]$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{R2}≔\mathrm{dec2}\left[2\right]$

$\mathrm{R2}≔\mathrm{polynomial\_ring}$

$\mathrm{Display}\left(\mathrm{rc2}\,\mathrm{R2}\right)$

$\left\{\begin{array}{cc}12x+5z^5+5=0& \\ 12y+5z^5+240z+5=0& \\ 3125z^{20}+12500z^{15}+18750z^{10}+12500z^5-59083=0& \end{array}\right.$

$\mathrm{dec0}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc2}\,\mathrm{R2}\,\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(M\right)\,\mathrm{vars}\right)$

$\mathrm{dec0}≔\left[\mathrm{regular\_chain}\,\mathrm{polynomial\_ring}\right]$

$\mathrm{rc0}≔\mathrm{dec0}\left[1\right]$

$\mathrm{rc0}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc0}\,R\right)$

$\left\{\begin{array}{cc}z+20x-y=0& \\ 5x^5+12y+5=0& \\ 5y^4-3=0& \end{array}\right.$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc}\,\mathrm{rc0}\,R\right)$

$\mathrm{true}$

$\mathrm{rc3}≔\mathrm{ChangeOfOrder}\left(\mathrm{rc}\,R\,\mathrm{R2}\right)$

$\mathrm{rc3}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc3}\,\mathrm{R2}\right)$

$\left\{\begin{array}{cc}12x+5z^5+5=0& \\ 12y+5z^5+240z+5=0& \\ 3125z^{20}+12500z^{15}+18750z^{10}+12500z^5-59083=0& \end{array}\right.$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc2}\,\mathrm{rc3}\,\mathrm{R2}\right)$

$\mathrm{true}$

$M≔\mathrm{Matrix}\left(\left[\left[0\,1\,0\right]\,\left[1\,0\,-1\right]\,\left[0\,0\,1\right]\right]\right)$

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

$\mathrm{vars2}≔\left[X\,Y\,Z\right]$

$\mathrm{vars2}≔\left[X\,Y\,Z\right]$

$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{dec}≔\mathrm{Triangularize}\left(F\,R\,'\mathrm{normalized}'='\mathrm{yes}'\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc}≔\mathrm{dec}\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}z+20x-y=0& \\ 5x^5+12y+5=0& \\ 5y^4-3=0& \end{array}\right.$

$\mathrm{dec2}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc}\,R\,M\,\mathrm{vars2}\right)$

$\mathrm{dec2}≔\left[\mathrm{regular\_chain}\,\mathrm{polynomial\_ring}\right]$

$\mathrm{rc2}≔\mathrm{dec2}\left[1\right]$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{R2}≔\mathrm{dec2}\left[2\right]$

$\mathrm{R2}≔\mathrm{polynomial\_ring}$

$\mathrm{Display}\left(\mathrm{rc2}\,\mathrm{R2}\right)$

$\left\{\begin{array}{cc}20X+Y-21Z=0& \\ 5Y^5-25Z{Y}^4+50Z^2{Y}^3-50Z^3{Y}^2+15Y-38400003Z-16000000=0& \\ 5Z^4-3=0& \end{array}\right.$

$\mathrm{dec0}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc2}\,\mathrm{R2}\,\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(M\right)\,\mathrm{vars}\right)$

$\mathrm{dec0}≔\left[\mathrm{regular\_chain}\,\mathrm{polynomial\_ring}\right]$

$\mathrm{rc0}≔\mathrm{dec0}\left[1\right]$

$\mathrm{rc0}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc0}\,R\right)$

$\left\{\begin{array}{cc}z+20x-y=0& \\ 5x^5+12y+5=0& \\ 5y^4-3=0& \end{array}\right.$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc}\,\mathrm{rc0}\,R\right)$

$\mathrm{true}$

Boulier, F.; Lemaire, F. and Moreno Maza, M. "PARDI!." Proc. ISSAC, 2001.

The RegularChains[ChainTools][ChangeOfCoordinates] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.