The command NormalizeRegularChain(rc, R) returns a triangular decomposition of rc into regular chains in the following sense. The intersection of the saturated ideals of the output regular chains has the same radical as the saturated ideal of rc. Moreover, each output regular chain is normalized.

In addition, if NormalizeRegularChain(rc, R) returns only one regular chain, then this regular chain has the same saturated ideal as rc.

More generally, if a saturated ideal $\mathrm{I}$ of an output regular chain has the same dimension as the saturated ideal $J$ of rc, then $\mathrm{I}$ is radical whenever $J$ is radical.

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

The notion of a (strongly) normalized regular chain is defined in the ChainTools subpackage overview page.

The notion of a saturated ideal (of a regular chain) is defined in the RegularChains package overview page.

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

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

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

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

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

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

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

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

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc}\,\mathrm{lrc}\left[1\right]\,R\right)$

$\mathrm{true}$

$\mathrm{IsStronglyNormalized}\left(\mathrm{lrc}\left[1\right]\,R\right)$

$\mathrm{true}$