The command Dimension(rc, R) returns the dimension of the saturated ideal of rc. This is also the number of variables of R minus the number of elements in rc.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

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

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

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

$\mathrm{map}\left(\mathrm{Dimension}\,\mathrm{decl}\,R\right)$

$\left[6\,5\,5\,4\,4\,4\,4\,3\,3\,3\,2\right]$

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left\{x\left(\left(x-1\right)\left(y-1\right)+\left(x-2\right)y\right)\,x\left(x-1\right)z\,x\left(x-1\right)\left(x-2\right)\right\}$

$\mathrm{sys}≔\left\{x\left(\left(x-1\right)\left(y-1\right)+\left(x-2\right)y\right)\,x\left(x-1\right)z\,x\left(x-1\right)\left(x-2\right)\right\}$

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

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

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

$\left[\left[x\right]\,\left[x-1\,y\right]\,\left[x-2\,y-1\,z\right]\right]$

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

$\left[2\,1\,0\right]$