GeneralConstruct - Maple Help

RegularChains[ConstructibleSetTools]

 GeneralConstruct
 construct a constructible set from a regular chain, equalities, and inequalities

 Calling Sequence GeneralConstruct(F, T, H, R) GeneralConstruct(F, T, R) GeneralConstruct(T, H, R) GeneralConstruct(F, H, R)

Parameters

 F, H - lists of polynomials T - regular chain R - polynomial ring

Description

 • The command GeneralConstruct(F, T, H, R) returns a constructible set $C$.
 Assume that the quasi-component of T is $W\left(T\right)$ (see RegularChains for the definition). Then $C$ consists of points in $W\left(T\right)$ which cancel all polynomials in F, but do not cancel any polynomials in H.
 • If F is not specified, it is set to be the empty list.
 • If T is not specified, it is set to be the empty regular chain.
 • If H is not specified, it is set to $\left[1\right]$.
 • The quasi-component of the empty regular chain is the whole space.
 • Any other inputs will be rejected and an error message will be reported.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form GeneralConstruct(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][GeneralConstruct](..).

Examples

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

First, define a polynomial ring and three polynomials in the ring.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $p≔\left(5t+5\right)x-y-\left(10t+7\right);$$q≔\left(5t-5\right)x-\left(t+2\right)y+\left(-7t+11\right);$$h≔x+t$
 ${p}{≔}\left({5}{}{t}{+}{5}\right){}{x}{-}{y}{-}{10}{}{t}{-}{7}$
 ${q}{≔}\left({5}{}{t}{-}{5}\right){}{x}{-}\left({t}{+}{2}\right){}{y}{-}{7}{}{t}{+}{11}$
 ${h}{≔}{x}{+}{t}$ (2)

Build a regular chain using $q$, which means $q$ vanishes but the initial $5t-5$ of $q$ does not vanish.

 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right):$$\mathrm{rc}≔\mathrm{Chain}\left(\left[q\right],\mathrm{rc},R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (3)

Use GeneralConstruct to figure out the points in $W\left(\mathrm{rc}\right)$ which cancel $p$ but do not cancel $h$.

 > $\mathrm{cs}≔\mathrm{GeneralConstruct}\left(\left[p\right],\mathrm{rc},\left[h\right],R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (4)

cs is a constructible set consisting of one regular system.

 > $\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs},R\right)$
 ${\mathrm{lrs}}{≔}\left[{\mathrm{regular_system}}\right]$ (5)

The inequalities form the following list.

 > $\mathrm{ineqs}≔\mathrm{map}\left(\mathrm{RepresentingInequations},\mathrm{lrs},R\right)$
 ${\mathrm{ineqs}}{≔}\left[\left[{t}{-}{1}{,}{{t}}^{{3}}{+}{4}{}{{t}}^{{2}}{+}{7}{}{t}{+}{5}\right]\right]$ (6)

To see complete information, use the Info command.

 > $\mathrm{Info}\left(\mathrm{cs},R\right)$
 $\left[\left[\left({5}{}{t}{-}{5}\right){}{x}{+}\left({-}{t}{-}{2}\right){}{y}{-}{7}{}{t}{+}{11}{,}\left({{t}}^{{2}}{+}{2}{}{t}{+}{3}\right){}{y}{-}{3}{}{{t}}^{{2}}{-}{t}{-}{4}\right]{,}\left[{t}{-}{1}{,}{{t}}^{{3}}{+}{4}{}{{t}}^{{2}}{+}{7}{}{t}{+}{5}\right]\right]$ (7)