ChangeOfCoordinates - Maple Help

RegularChains[ChainTools]

 ChangeOfCoordinates
 change of coordinate system for a regular chain

 Calling Sequence ChangeOfCoordinates(rc, R, M, V)

Parameters

 rc - regular chain of R R - polynomial ring M - matrix V - variable list

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $F≔\left[5{y}^{4}-3,20x-y+z,{x}^{5}-{y}^{5}+3y+1\right]$
 ${F}{≔}\left[{5}{}{{y}}^{{4}}{-}{3}{,}{20}{}{x}{-}{y}{+}{z}{,}{{x}}^{{5}}{-}{{y}}^{{5}}{+}{3}{}{y}{+}{1}\right]$ (1)
 > $\mathrm{vars}≔\left[z,x,y\right]$
 ${\mathrm{vars}}{≔}\left[{z}{,}{x}{,}{y}\right]$ (2)

In the example below, we consider a change of coordinates which is given by a permutation matrix, hence, this example is a change of variable order.

 > $\mathrm{vars2}≔\left[x,y,z\right]$
 ${\mathrm{vars2}}{≔}\left[{x}{,}{y}{,}{z}\right]$ (3)
 > $M≔\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[0,0,1\right],\left[1,0,0\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{0}& {1}& {0}\\ {0}& {0}& {1}\\ {1}& {0}& {0}\end{array}\right]$ (4)
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (5)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,'\mathrm{normalized}'='\mathrm{yes}'\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$ (6)
 > $\mathrm{rc}≔\mathrm{dec}\left[1\right]$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (7)
 > $\mathrm{Display}\left(\mathrm{rc},R\right)$
 $\left\{\begin{array}{cc}{z}{+}{20}{}{x}{-}{y}{=}{0}& {}\\ {5}{}{{x}}^{{5}}{+}{12}{}{y}{+}{5}{=}{0}& {}\\ {5}{}{{y}}^{{4}}{-}{3}{=}{0}& {}\end{array}\right\$ (8)
 > $\mathrm{dec2}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc},R,M,\mathrm{vars2}\right)$
 ${\mathrm{dec2}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{polynomial_ring}}\right]$ (9)
 > $\mathrm{rc2}≔\mathrm{dec2}\left[1\right]$
 ${\mathrm{rc2}}{≔}{\mathrm{regular_chain}}$ (10)
 > $\mathrm{R2}≔\mathrm{dec2}\left[2\right]$
 ${\mathrm{R2}}{≔}{\mathrm{polynomial_ring}}$ (11)
 > $\mathrm{Display}\left(\mathrm{rc2},\mathrm{R2}\right)$
 $\left\{\begin{array}{cc}{12}{}{x}{+}{5}{}{{z}}^{{5}}{+}{5}{=}{0}& {}\\ {12}{}{y}{+}{5}{}{{z}}^{{5}}{+}{240}{}{z}{+}{5}{=}{0}& {}\\ {3125}{}{{z}}^{{20}}{+}{12500}{}{{z}}^{{15}}{+}{18750}{}{{z}}^{{10}}{+}{12500}{}{{z}}^{{5}}{-}{59083}{=}{0}& {}\end{array}\right\$ (12)
 > $\mathrm{dec0}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc2},\mathrm{R2},\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(M\right),\mathrm{vars}\right)$
 ${\mathrm{dec0}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{polynomial_ring}}\right]$ (13)
 > $\mathrm{rc0}≔\mathrm{dec0}\left[1\right]$
 ${\mathrm{rc0}}{≔}{\mathrm{regular_chain}}$ (14)
 > $\mathrm{Display}\left(\mathrm{rc0},R\right)$
 $\left\{\begin{array}{cc}{z}{+}{20}{}{x}{-}{y}{=}{0}& {}\\ {5}{}{{x}}^{{5}}{+}{12}{}{y}{+}{5}{=}{0}& {}\\ {5}{}{{y}}^{{4}}{-}{3}{=}{0}& {}\end{array}\right\$ (15)
 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc},\mathrm{rc0},R\right)$
 ${\mathrm{true}}$ (16)

Because this change of coordinates is a change of variable order, it can also be performed by the ChangeOfOrder command.

 > $\mathrm{rc3}≔\mathrm{ChangeOfOrder}\left(\mathrm{rc},R,\mathrm{R2}\right)$
 ${\mathrm{rc3}}{≔}{\mathrm{regular_chain}}$ (17)
 > $\mathrm{Display}\left(\mathrm{rc3},\mathrm{R2}\right)$
 $\left\{\begin{array}{cc}{12}{}{x}{+}{5}{}{{z}}^{{5}}{+}{5}{=}{0}& {}\\ {12}{}{y}{+}{5}{}{{z}}^{{5}}{+}{240}{}{z}{+}{5}{=}{0}& {}\\ {3125}{}{{z}}^{{20}}{+}{12500}{}{{z}}^{{15}}{+}{18750}{}{{z}}^{{10}}{+}{12500}{}{{z}}^{{5}}{-}{59083}{=}{0}& {}\end{array}\right\$ (18)
 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc2},\mathrm{rc3},\mathrm{R2}\right)$
 ${\mathrm{true}}$ (19)

In the next example, we consider a change of coordinates which is not a change of variable order.

 > $M≔\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[1,0,-1\right],\left[0,0,1\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{0}& {1}& {0}\\ {1}& {0}& {-1}\\ {0}& {0}& {1}\end{array}\right]$ (20)
 > $\mathrm{vars2}≔\left[X,Y,Z\right]$
 ${\mathrm{vars2}}{≔}\left[{X}{,}{Y}{,}{Z}\right]$ (21)
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (22)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,'\mathrm{normalized}'='\mathrm{yes}'\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$ (23)
 > $\mathrm{rc}≔\mathrm{dec}\left[1\right]$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (24)
 > $\mathrm{Display}\left(\mathrm{rc},R\right)$
 $\left\{\begin{array}{cc}{z}{+}{20}{}{x}{-}{y}{=}{0}& {}\\ {5}{}{{x}}^{{5}}{+}{12}{}{y}{+}{5}{=}{0}& {}\\ {5}{}{{y}}^{{4}}{-}{3}{=}{0}& {}\end{array}\right\$ (25)
 > $\mathrm{dec2}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc},R,M,\mathrm{vars2}\right)$
 ${\mathrm{dec2}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{polynomial_ring}}\right]$ (26)
 > $\mathrm{rc2}≔\mathrm{dec2}\left[1\right]$
 ${\mathrm{rc2}}{≔}{\mathrm{regular_chain}}$ (27)
 > $\mathrm{R2}≔\mathrm{dec2}\left[2\right]$
 ${\mathrm{R2}}{≔}{\mathrm{polynomial_ring}}$ (28)
 > $\mathrm{Display}\left(\mathrm{rc2},\mathrm{R2}\right)$
 $\left\{\begin{array}{cc}{20}{}{X}{+}{Y}{-}{21}{}{Z}{=}{0}& {}\\ {5}{}{{Y}}^{{5}}{-}{25}{}{Z}{}{{Y}}^{{4}}{+}{50}{}{{Z}}^{{2}}{}{{Y}}^{{3}}{-}{50}{}{{Z}}^{{3}}{}{{Y}}^{{2}}{+}{15}{}{Y}{-}{38400003}{}{Z}{-}{16000000}{=}{0}& {}\\ {5}{}{{Z}}^{{4}}{-}{3}{=}{0}& {}\end{array}\right\$ (29)
 > $\mathrm{dec0}≔\mathrm{ChangeOfCoordinates}\left(\mathrm{rc2},\mathrm{R2},\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(M\right),\mathrm{vars}\right)$
 ${\mathrm{dec0}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{polynomial_ring}}\right]$ (30)
 > $\mathrm{rc0}≔\mathrm{dec0}\left[1\right]$
 ${\mathrm{rc0}}{≔}{\mathrm{regular_chain}}$ (31)
 > $\mathrm{Display}\left(\mathrm{rc0},R\right)$
 $\left\{\begin{array}{cc}{z}{+}{20}{}{x}{-}{y}{=}{0}& {}\\ {5}{}{{x}}^{{5}}{+}{12}{}{y}{+}{5}{=}{0}& {}\\ {5}{}{{y}}^{{4}}{-}{3}{=}{0}& {}\end{array}\right\$ (32)
 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc},\mathrm{rc0},R\right)$
 ${\mathrm{true}}$ (33)

References

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

Compatibility

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