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.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{IsEmpty}\left(\mathrm{cs}\,R\right)$

$\mathrm{true}$

$\mathrm{IsEmpty}\left(\left[x^2+y^2+1=0\right]\,R\right)$

$\mathrm{true}$

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

$\mathrm{false}$

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.