IsEmpty - Maple Help

RegularChains

 ConstructibleSetTools[IsEmpty]
 check whether a constructible set is empty or not
 SemiAlgebraicSetTools[IsEmpty]
 check whether a semi-algebraic set is empty or not

 Calling Sequence IsEmpty(cs, R) IsEmpty(sys, R) IsEmpty(F,N,P,H, R)

Parameters

 cs - constructible set structure sys - list of polynomial constraints F - list of polynomials N - list of polynomials P - list of polynomials H - list of polynomials R - polynomial ring

Description

 • 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 semi-algebraic 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.

Examples

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

Define a polynomial ring $R$ first.

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

Then construct a constructible set cs as follows.

 > $\mathrm{cs}≔\mathrm{GeneralConstruct}\left(\left[{x}^{2},xy-1\right],\left[1\right],R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (2)
 > $\mathrm{IsEmpty}\left(\mathrm{cs},R\right)$
 ${\mathrm{true}}$ (3)

Semi-algebraic case:

 > $\mathrm{IsEmpty}\left(\left[{x}^{2}+{y}^{2}+1=0\right],R\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsEmpty}\left(\left[{x}^{2}+{y}^{2}-1=0\right],R\right)$
 ${\mathrm{false}}$ (5)

Compatibility

 • The RegularChains[SemiAlgebraicSetTools][IsEmpty] command was introduced in Maple 16.
 • The sys, F, N, P, H and R parameters were introduced in Maple 16.