The command Squarefree(rc, R) returns a triangular decomposition of rc into regular chains with square-free saturated ideals. This triangular decomposition is the sense of Kalkbrener, that is, the radical of the the saturated ideal of rc must equal the intersection of the radical ideals of the saturated ideals of the output regular chains.

If 'normalized'='yes' is provided, then the output regular chains are also normalized.

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

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

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\left[x^2-2yx+z\,y^2-z\right]$

$\mathrm{lrc}≔\mathrm{Squarefree}\left(\mathrm{rc}\,R\right)$

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

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

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