The command MakePairwiseDisjoint(cs, R) returns a constructible set cs1 such that cs1 and cs are equal and the regular systems representing cs1 are pairwise disjoint.

Generally, in a constructible set, there is some redundancy among its components defined by regular systems. By default, functions on constructible sets do not remove redundancy because such a computation is expensive.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔ax\+by-e$

$F≔ax+by-e$

$G≔cx\+dy-e$

$G≔cx+dy-e$

$\mathrm{cs}≔\mathrm{Projection}\left(\left[F\,G\right]\,5\,R\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs}\,R\right)$

$\mathrm{lrs}≔\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\right]$

$\mathrm{Info}\left(\mathrm{cs}\,R\right)$

$\left[\left[\right]\,\left[c\,da-bc\right]\right],\left[\left[a-c\,b-d\right]\,\left[c\right]\right],\left[\left[c\right]\,\left[d\,a\right]\right],\left[\left[da-bc\,e\right]\,\left[d\,c\right]\right],\left[\left[a\,b-d\,c\right]\,\left[d\right]\right],\left[\left[a\,c\,e\right]\,\left[1\right]\right],\left[\left[b\,d\,e\right]\,\left[1\right]\right],\left[\left[c\,d\,e\right]\,\left[a\right]\right],\left[\left[a\,b\,c\,d\,e\right]\,\left[1\right]\right]$

$\mathrm{nops}\left(\mathrm{lrs}\right)$

$9$

$\mathrm{cs\_mpd}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs}\,R\right)$

$\mathrm{cs\_mpd}≔\mathrm{constructible\_set}$

$\mathrm{lcs\_mpd}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs\_mpd}\,R\right)$

$\mathrm{lcs\_mpd}≔\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\right]$

$\mathrm{nops}\left(\mathrm{lcs\_mpd}\right)$

$9$

$\mathrm{Info}\left(\mathrm{cs\_mpd}\,R\right)$

$\left[\left[a\,b\,c\,d\,e\right]\,\left[1\right]\right],\left[\left[c\,d\,e\right]\,\left[a\,b\right]\right],\left[\left[b\,d\,e\right]\,\left[a-c\right]\right],\left[\left[a\,c\,e\right]\,\left[b-d\right]\right],\left[\left[da-bc\,e\right]\,\left[d\,c\,b-d\right]\right],\left[\left[a\,b-d\,c\right]\,\left[d\right]\right],\left[\left[c\right]\,\left[d\,a\right]\right],\left[\left[a-c\,b-d\right]\,\left[c\right]\right],\left[\left[\right]\,\left[c\,da-bc\right]\right]$