The command Construct(p, rc, R) returns a list of regular chains ${\mathrm{rc}}_i$ which form a triangular decomposition of the regular chain obtained by extending rc with p.

This assumes that p is a non-constant with main variable greater than any algebraic variable of rc, and that the initial of p is regular modulo the saturated ideal of rc. Hence p and rc form together a regular chain.

Although rc with p is assumed to form a regular chain, several regular chains may be returned; this is because the polynomial p may be factorized with respect to rc in order to simplify the expressions in the regular chains ${\mathrm{rc}}_i$.

Such factorizations will happen if they can be performed quickly. For instance, if p involves only one variable.

To avoid these possible factorizations, use RegularChains[ChainTools][Chain]

If 'normalized'='yes' is present, then rc must be normalized. In addition, every returned regular chain is normalized.

If 'normalized'='strongly' is present, then rc must be strongly normalized. In addition, every returned regular chain is strongly normalized.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{pz}≔z^2+2z+1$

$\mathrm{pz}≔z^2+2z+1$

$\mathrm{py}≔y^2+z$

$\mathrm{py}≔y^2+z$

$\mathrm{pt}≔t^3+yz$

$\mathrm{pt}≔t^3+yz$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc1}≔\mathrm{Construct}\left(\mathrm{pz}\,\mathrm{rc}\,R\right)$

$\mathrm{rc1}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc1}≔\mathrm{rc1}\left[1\right]\;$$\mathrm{Equations}\left(\mathrm{rc1}\,R\right)$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\left[z+1\right]$

$\mathrm{rc2}≔\mathrm{Construct}\left(\mathrm{py}\,\mathrm{rc1}\,R\right)$

$\mathrm{rc2}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{rc2}≔\mathrm{rc2}\left[1\right]\;$$\mathrm{Equations}\left(\mathrm{rc2}\,R\right)$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\left[y-1\,z+1\right]$

$\mathrm{rc3}≔\mathrm{Construct}\left(\mathrm{pt}\,\mathrm{rc2}\,R\right)$

$\mathrm{rc3}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{rc3}\,R\right)$

$\left[\left[t-1\,y-1\,z+1\right]\,\left[t^2+t+1\,y-1\,z+1\right]\right]$