The command ChangeOfOrder returns a regular chain rc2 of R2 which has the same saturated ideal as rc.

The saturated ideal of the regular chain rc must be universally characterizable; in particular, the algorithm applies if the saturated ideal of rc is a prime ideal.

The key point of this approach is to reduce to the dimension zero case, that is, isolating this particular case as the central one. In order to achieve this reduction, the input regular chain is transformed by a sequence of elementary changes of variable orders in dimension zero. Matroid theory is used to compute this sequence of intermediate variable orders.  Several known techniques, notably lifting techniques and rational reconstruction, play important roles here.

This command is part of the ChainTools package, so it can be used in the form ChangeOfOrder(..) 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][ChangeOfOrder](..).

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

$R≔\mathrm{PolynomialRing}\left(\left[\mathrm{P1}\,\mathrm{P2}\,S\,\mathrm{X2}\,\mathrm{X1}\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{R2}≔\mathrm{PolynomialRing}\left(\left[\mathrm{X1}\,\mathrm{X2}\,S\,\mathrm{P2}\,\mathrm{P1}\right]\right)$

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

$F≔\left[\mathrm{P1}-{\mathrm{X1}}^2\,\mathrm{P2}-{\mathrm{X2}}^2\,S-\mathrm{X1}\mathrm{X2}\right]$

$F≔\left[-{\mathrm{X1}}^2+\mathrm{P1}\,-{\mathrm{X2}}^2+\mathrm{P2}\,-\mathrm{X1}\mathrm{X2}+S\right]$

$\mathrm{rc}≔\mathrm{Triangularize}\left(F\,R\,\mathrm{normalized}=\mathrm{yes}\right)\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc2}≔\mathrm{ChangeOfOrder}\left(\mathrm{rc}\,R\,\mathrm{R2}\right)$

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

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

$\left[\mathrm{X2}\mathrm{X1}-S\,{\mathrm{X2}}^2-\mathrm{P2}\,S^2-\mathrm{P2}\mathrm{P1}\right]$

$\mathrm{rc1}≔\mathrm{ChangeOfOrder}\left(\mathrm{rc2}\,\mathrm{R2}\,R\right)$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{rc1}\,R\right)$

$\left[\mathrm{P1}-{\mathrm{X1}}^2\,\mathrm{P2}-{\mathrm{X2}}^2\,S-\mathrm{X2}\mathrm{X1}\right]$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc}\,\mathrm{rc1}\,R\right)$

$\mathrm{true}$

Dahan, X.; Jin, X.; Moreno Maza, M. and Schost, E. "Change of Ordering for Regular Chains in Positive Dimension." Proc. Maple Conference, Waterloo, 2006.