The command Complement(cs, R) returns the complement of the constructible set cs in the affine space associated with R. If K is the algebraic closure of the coefficient field of R and n is the number of variables in R, then this affine space is $K^n$.  The polynomial ring may have characteristic zero or a prime characteristic.

The command Complement(lrsas, R) returns the complement of the semi-algebraic set represented by lrsas (see RealTriangularize for this representation). The polynomial ring must have characteristic zero. The empty semi-algebraic set is encoded by the empty list.

The empty constructible set represents the empty set of $K^n$.

This command is available once RegularChains[ConstructibleSetTools] submodule or RegularChains[SemiAlgebraicSetTools] submodule have been loaded. It can always be accessed through one of the following long forms: RegularChains[ConstructibleSetTools][Complement] or RegularChains[SemiAlgebraicSetTools][Complement].

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

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,a\,b\,c\,d\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔ax+c$

$F≔ax+c$

$G≔bx+d$

$G≔bx+d$

$\mathrm{cs}≔\mathrm{Projection}\left(\left[F\,G\right]\,4\,R\right)$

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

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

$\left[\left[da-cb\right]\,\left[d\,b\right]\right],\left[\left[b\,d\right]\,\left[a\right]\right],\left[\left[c\,d\right]\,\left[1\right]\right],\left[\left[a\,b\,c\,d\right]\,\left[1\right]\right]$

$\mathrm{com\_cs}≔\mathrm{Complement}\left(\mathrm{cs}\,R\right)$

$\mathrm{com\_cs}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{com\_cs}\,R\right)$

$\left[\left[\right]\,\left[da-bc\,d\right]\right],\left[\left[d\right]\,\left[c\,b\right]\right],\left[\left[a\,b\right]\,\left[d\right]\right],\left[\left[a\,b\,d\right]\,\left[c\right]\right]$

$\mathrm{com\_com\_cs}≔\mathrm{Complement}\left(\mathrm{com\_cs}\,R\right)$

$\mathrm{com\_com\_cs}≔\mathrm{constructible\_set}$

$\mathrm{IsContained}\left(\mathrm{cs}\,\mathrm{com\_com\_cs}\,R\right)\mathbf{and}\mathrm{IsContained}\left(\mathrm{com\_com\_cs}\,\mathrm{cs}\,R\right)$

$\mathrm{true}$

$R≔\mathrm{PolynomialRing}\left(\left[a\,b\,c\right]\right)\:$

$\mathrm{S1}≔\left[a^2-ca-a=0\&comma;0<a-c\right]$

$\mathrm{S1}≔\left[a^2-ca-a=0\&comma;0<a-c\right]$

$\mathrm{out}≔\mathrm{RealTriangularize}\left(\mathrm{S1}\&comma;R\right)$

$\mathrm{out}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{compl}≔\mathrm{Complement}\left(\mathrm{out}\&comma;R\right)$

$\mathrm{compl}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{expected}≔\left[\left[a^2-ca-a\ne 0\right]\&comma;\left[a-c\le 0\right]\right]$

$\mathrm{expected}≔\left[\left[a^2-ca-a\ne 0\right]\&comma;\left[a-c\le 0\right]\right]$

$\mathrm{expected}≔\mathrm{map}\left(t\mapsto \mathrm{op}\left(\mathrm{RealTriangularize}\left(t\&comma;R\right)\right)\&comma;\mathrm{expected}\right)$

$\mathrm{expected}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Difference}\left(\mathrm{compl}\&comma;\mathrm{expected}\&comma;R\right)$

$\mathrm{Difference}\left(\mathrm{expected}\&comma;\mathrm{compl}\&comma;R\right)$

Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition". Proc. CASC 2007, LNCS, Vol. 4770: 73-101. Springer, 2007.

Chen, C.; Davenport, J.-D.; Moreno Maza, M.; Xia, B.; and Xiao, R. "Computing with semi-algebraic sets represented by triangular decomposition". Proceedings of 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), ACM Press, pp. 75--82, 2011.

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

The lrsas parameter was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.