RegularChains - Maple Programming Help

Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : RegularChains/ConstructibleSetTools/EmptyConstructibleSet

RegularChains

 ConstructibleSetTools[EmptyConstructibleSet]
 construct the empty constructible set
 SemiAlgebraicSetTools[EmptySemiAlgebraicSet]
 construct the empty semi-algebraic set

 Calling Sequence EmptyConstructibleSet(R) EmptySemiAlgebraicSet(R)

Parameters

 R - polynomial ring

Description

 • The command EmptyConstructibleSet(R) returns the empty set of the affine space over the algebraic closure of the base field of R and whose dimension is equal to the total number of variables of R.
 • This command is part of the RegularChains[ConstructibleSetTools] submodule, so it can be used in the form EmptyConstructibleSet(..) 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][EmptyConstructibleSet](..).
 • The command EmptySemiAlgebraicSet(R) returns the empty set of the euclidean space over the real numbers and whose dimension is equal to the total number of variables of R.
 • This command is part of the RegularChains[SemiAlgebraicSetTools] submodule, so it can be used in the form EmptySemiAlgebraicSet(..) only after executing the command with(RegularChains[SemiAlgebraicSetTools].  However, it can always be accessed through the long form of the command by using RegularChains[SemiAlgebraicSetTools][EmptySemiAlgebraicSet](..).

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)

Then construct an empty constructible set cs as follows.

 > $\mathrm{cs}≔\mathrm{EmptyConstructibleSet}\left(R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (2)
 > $\mathrm{IsEmpty}\left(\mathrm{cs},R\right)$
 ${\mathrm{true}}$ (3)

Compatibility

 • The RegularChains[SemiAlgebraicSetTools][EmptySemiAlgebraicSet] command was introduced in Maple 16.