RegularChains[ConstructibleSetTools] - Maple Programming Help

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RegularChains[ConstructibleSetTools]

 Union
 compute the union of two constructible sets

 Calling Sequence Union(cs1, cs2, R)

Parameters

 cs1, cs2 - constructible sets R - polynomial ring

Description

 • The command Union(cs1, cs2, R) returns a constructible set, the union of cs1 and cs2.
 • There might be redundancy in the output.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form Union(..) 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][Union](..).

Examples

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

Define a polynomial ring $R$ first.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Consider the following polynomials of $R$.

 > $F≔2{x}^{2}+3xy+{y}^{2}-3x-3y$
 ${F}{≔}{2}{}{{x}}^{{2}}{+}{3}{}{x}{}{y}{+}{{y}}^{{2}}{-}{3}{}{x}{-}{3}{}{y}$ (2)
 > $G≔x{y}^{2}-x$
 ${G}{≔}{x}{}{{y}}^{{2}}{-}{x}$ (3)
 > $H≔{y}^{3}-y$
 ${H}{≔}{{y}}^{{3}}{-}{y}$ (4)

Let cs1 be the solution set of $F$ and $G$, and cs2 be the solution set of $G$ and $H$.

 > $\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(\left[F,G\right],\left[1\right],R\right)$
 ${\mathrm{cs1}}{≔}{\mathrm{constructible_set}}$ (5)
 > $\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[G,H\right],\left[1\right],R\right)$
 ${\mathrm{cs2}}{≔}{\mathrm{constructible_set}}$ (6)

Use the command Union to obtain the union of these two solution sets.

 > $\mathrm{cs3}≔\mathrm{Union}\left(\mathrm{cs1},\mathrm{cs2},R\right):$$\mathrm{Info}\left(\mathrm{cs3},R\right)$
 $\left[\left[{x}{-}{2}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{+}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}{-}{3}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]$
 $\left\{\left[\left[{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}{-}{3}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{2}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{-}{1}{,}{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{+}{1}{,}{y}{-}{1}\right]{,}\left[{1}\right]\right]\right\}$ (7)

The Union command is not guaranteed to remove all the redundant components, for efficiency consideration.  Use the MakePairwiseDisjoint command to remove them.

 > $\mathrm{cs3}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs3},R\right)$
 ${\mathrm{cs3}}{≔}{\mathrm{constructible_set}}$ (8)
 > $\mathrm{Info}\left(\mathrm{cs3},R\right)$
 $\left[\left[{x}{,}{y}{-}{3}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{-}{1}\right]{,}\left[{1}\right]\right]$
 $\left\{\left[\left[{y}{-}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{y}{+}{1}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}\right]{,}\left[{1}\right]\right]{,}\left[\left[{x}{,}{y}{-}{3}\right]{,}\left[{1}\right]\right]\right\}$ (9)