The command Extend(rc, lp, R) returns a triangular decomposition (by means of regular chains) of the quasi-component defined by rc and lp. This assumes that polynomials of lp form a triangular set and are sorted in an ascending order according to their main variables. Moreover, it is assumed that each main variable of a polynomial in lp is larger than any variable appearing in rc. Therefore, the polynomials in rc and lp together must form a triangular set, which is, however, not necessarily a regular chain.

If the option 'output'='lazard' is present then the triangular decomposition is the sense of Lazard otherwise it is in the sense of Kalkbrener.

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

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

$R≔\mathrm{polynomial\_ring}$

$C≔\mathrm{Chain}\left(\left[y^2-x^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$C≔\mathrm{regular\_chain}$

$E≔\mathrm{Extend}\left(C\,\left[\left(y-x\right)z^2+\left(y+x\right)z\right]\,R\right)\;$$\mathrm{map}\left(\mathrm{Display}\,E\,R\right)$

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

$\left[\left\{\begin{array}{cc}z=0& \\ y+x=0& \end{array}\right.\right]$

$E≔\mathrm{Extend}\left(C\,\left[\left(y-x\right)z^2+z\right]\,R\right)\;$$\mathrm{map}\left(\mathrm{Display}\,E\,R\right)$

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

$\left[\left\{\begin{array}{cc}z=0& \\ y+x=0& \end{array}\right.\,\left\{\begin{array}{cc}2xz-1=0& \\ y+x=0& \\ 2x\ne 0& \end{array}\right.\right]$

The RegularChains[ChainTools][Extend] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.