The command Cut(v, rc, R) returns $\left[\mathrm{alg\_v}\,\mathrm{top}\,\mathrm{pv}\,\mathrm{sub\_rc}\right]$ where $\mathrm{alg\_v}$ is true if and only if v is an algebraic variable of rc, that is, if there exists a polynomial of rc with v as main variables.

Moreover, top is assigned to the list of polynomials of rc with main variable strictly greater than v, ordered by increasing main variable.

pv is assigned to the polynomial of rc with main variable v. If no such polynomial exists, pv is left unchanged.

sub_rc is the regular chain under the variable v, i.e. composed of the polynomials of rc with main variable strictly lower than v.

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z+1\,y+2\,x+3\right]\,\mathrm{rc}\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{res}≔\mathrm{Cut}\left(x\,\mathrm{rc}\,R\right)\;$$\mathrm{Equations}\left(\mathrm{res}\left[4\right]\,R\right)$

$\mathrm{res}≔\left[\mathrm{true}\,\left[\right]\,x+3\,\mathrm{regular\_chain}\right]$

$\left[y+2\,z+1\right]$

$\mathrm{res}≔\mathrm{Cut}\left(y\,\mathrm{rc}\,R\right)\;$$\mathrm{Equations}\left(\mathrm{res}\left[4\right]\,R\right)$

$\mathrm{res}≔\left[\mathrm{true}\,\left[x+3\right]\,y+2\,\mathrm{regular\_chain}\right]$

$\left[z+1\right]$

$\mathrm{res}≔\mathrm{Cut}\left(z\,\mathrm{rc}\,R\right)\;$$\mathrm{Equations}\left(\mathrm{res}\left[4\right]\,R\right)$

$\mathrm{res}≔\left[\mathrm{true}\,\left[y+2\,x+3\right]\,z+1\,\mathrm{regular\_chain}\right]$