RegularChains
ConstructibleSetTools[IsEmpty]
check whether a constructible set is empty or not
SemiAlgebraicSetTools[IsEmpty]
check whether a semialgebraic set is empty or not
Calling Sequence
Parameters
Description
Examples
Compatibility
IsEmpty(cs, R)
IsEmpty(sys, R)
IsEmpty(F,N,P,H, R)
cs

constructible set structure
sys
list of polynomial constraints
F
list of polynomials
N
P
H
R
polynomial ring
The command IsEmpty(cs, R) decides whether the constructible set cs is empty or not. The polynomial ring may have characteristic zero or a prime characteristic.
The command IsEmpty(sys, R) checks whether the zero set of the set of constraints sys is empty or not. The constraints in sys can be any polynomial equations, inequations or inequalities given by polynomials of R. The zero set of sys is regarded as the set of the real solutions of the polynomial system defined by sys. This assumes that R has characteristic zero.
The command IsEmpty(F, N, P, H, R) checks whether the zero set of the semialgebraic system represented by [F, N, P, H] (see RealTriangularize) is empty or not. This assumes that R has characteristic zero.
This command is available once either the RegularChains[ConstructibleSetTools] submodule or the RegularChains[SemiAlgebraicSetTools] submodule have been loaded. It can always be accessed through one of the following long forms: RegularChains:ConstructibleSetTools:IsEmpty or RegularChains:SemiAlgebraicSetTools:IsEmpty.
$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$
$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$
$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$
Define a polynomial ring $R$ first.
$R\u2254\mathrm{PolynomialRing}\left(\left[x\,y\right]\right)$
${R}{\u2254}{\mathrm{polynomial\_ring}}$
Then construct a constructible set cs as follows.
$\mathrm{cs}\u2254\mathrm{GeneralConstruct}\left(\left[{x}^{2}\,xy1\right]\,\left[1\right]\,R\right)$
${\mathrm{cs}}{\u2254}{\mathrm{constructible\_set}}$
$\mathrm{IsEmpty}\left(\mathrm{cs}\,R\right)$
${\mathrm{true}}$
Semialgebraic case:
$\mathrm{IsEmpty}\left(\left[{x}^{2}+{y}^{2}+1=0\right]\,R\right)$
$\mathrm{IsEmpty}\left(\left[{x}^{2}+{y}^{2}1=0\right]\,R\right)$
${\mathrm{false}}$
The RegularChains[SemiAlgebraicSetTools][IsEmpty] command was introduced in Maple 16.
The sys, F, N, P, H and R parameters were introduced in Maple 16.
For more information on Maple 16 changes, see Updates in Maple 16.
See Also
ConstructibleSet
ConstructibleSetTools
EmptyConstructibleSet
EmptySemiAlgebraicSet
GeneralConstruct
IsContained
RealTriangularize
SemiAlgebraicSetTools
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