The command Inequations(rc,R) returns the set of the initials of rc.

By definition, a zero of the regular chain rc is a common zero of its equations that does not cancel any of the initials of rc.

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,a\,b\,c\,d\,g\,h\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left\{ax+by-g\,cx+dy-h\right\}$

$\mathrm{sys}≔\left\{ax+by-g\,cx+dy-h\right\}$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\right)\;$$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

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

$\left[\left[cx+yd-h\,\left(ad-bc\right)y-ah+cg\right]\right]$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{output}=\mathrm{lazard}\right)$

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

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

$\left[\left[cx+dy-h\,\left(da-bc\right)y-ha+cg\right]\,\left[cx+dy-h\,da-bc\,hb-dg\right]\,\left[ax+by-g\,dy-h\,c\right]\,\left[dy-h\,a\,hb-dg\,c\right]\,\left[cx-h\,ha-cg\,b\,d\right]\,\left[ax+by-g\,c\,d\,h\right]\,\left[cx+dy\,da-bc\,g\,h\right]\,\left[by-g\,a\,c\,d\,h\right]\,\left[y\,a\,c\,g\,h\right]\,\left[x\,b\,d\,g\,h\right]\,\left[a\,b\,c\,d\,g\,h\right]\right]$

$\mathrm{map}\left(\mathrm{Inequations}\,\mathrm{dec}\,R\right)$

$\left[\left\{c\,da-bc\right\}\,\left\{c\,d\,h\right\}\,\left\{a\,d\right\}\,\left\{d\,h\right\}\,\left\{c\,h\right\}\,\left\{a\right\}\,\left\{c\,d\right\}\,\left\{b\right\}\,\varnothing \,\varnothing \,\varnothing \right]$

$\left[\mathrm{seq}\left(\left[\mathrm{eq}=\mathrm{Equations}\left(\mathrm{dec}\left[i\right]\,R\right)\,\mathrm{ineq}=\mathrm{Inequations}\left(\mathrm{dec}\left[i\right]\,R\right)\right]\,i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)\right]$

$\left[\left[\mathrm{eq}=\left[cx+dy-h\,\left(da-bc\right)y-ha+cg\right]\,\mathrm{ineq}=\left\{c\,da-bc\right\}\right]\,\left[\mathrm{eq}=\left[cx+dy-h\,da-bc\,hb-dg\right]\,\mathrm{ineq}=\left\{c\,d\,h\right\}\right]\,\left[\mathrm{eq}=\left[ax+by-g\,dy-h\,c\right]\,\mathrm{ineq}=\left\{a\,d\right\}\right]\,\left[\mathrm{eq}=\left[dy-h\,a\,hb-dg\,c\right]\,\mathrm{ineq}=\left\{d\,h\right\}\right]\,\left[\mathrm{eq}=\left[cx-h\,ha-cg\,b\,d\right]\,\mathrm{ineq}=\left\{c\,h\right\}\right]\,\left[\mathrm{eq}=\left[ax+by-g\,c\,d\,h\right]\,\mathrm{ineq}=\left\{a\right\}\right]\,\left[\mathrm{eq}=\left[cx+dy\,da-bc\,g\,h\right]\,\mathrm{ineq}=\left\{c\,d\right\}\right]\,\left[\mathrm{eq}=\left[by-g\,a\,c\,d\,h\right]\,\mathrm{ineq}=\left\{b\right\}\right]\,\left[\mathrm{eq}=\left[y\,a\,c\,g\,h\right]\,\mathrm{ineq}=\varnothing \right]\,\left[\mathrm{eq}=\left[x\,b\,d\,g\,h\right]\,\mathrm{ineq}=\varnothing \right]\,\left[\mathrm{eq}=\left[a\,b\,c\,d\,g\,h\right]\,\mathrm{ineq}=\varnothing \right]\right]$