Extend - Maple Help

RegularChains[ChainTools]

 Extend
 decomposes a triangular set into regular chains

 Calling Sequence Extend(rc, lp, R) Extend(rc, lp, R,  'output'='lazard')

Parameters

 rc - regular chain of R lp - polynomial of R R - polynomial ring 'output'='lazard' - (optional) boolean flag

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $C≔\mathrm{Chain}\left(\left[{y}^{2}-{x}^{2}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${C}{≔}{\mathrm{regular_chain}}$ (2)
 > $E≔\mathrm{Extend}\left(C,\left[\left(y-x\right){z}^{2}+\left(y+x\right)z\right],R\right);$$\mathrm{map}\left(\mathrm{Display},E,R\right)$
 ${E}{≔}\left[{\mathrm{regular_chain}}\right]$
 $\left[\left\{\begin{array}{cc}{z}{=}{0}& {}\\ {y}{+}{x}{=}{0}& {}\end{array}\right\\right]$ (3)
 > $E≔\mathrm{Extend}\left(C,\left[\left(y-x\right){z}^{2}+z\right],R\right);$$\mathrm{map}\left(\mathrm{Display},E,R\right)$
 ${E}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$
 $\left[\left\{\begin{array}{cc}{z}{=}{0}& {}\\ {y}{+}{x}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{2}{}{x}{}{z}{-}{1}{=}{0}& {}\\ {y}{+}{x}{=}{0}& {}\\ {2}{}{x}{\ne }{0}& {}\end{array}\right\\right]$ (4)

Compatibility

 • The RegularChains[ChainTools][Extend] command was introduced in Maple 15.
 • For more information on Maple 15 changes, see Updates in Maple 15.