This command computes the set-theoretic difference of two constructible sets or two semi-algebraic sets, depending on the input type.

A constructible set must be encoded as an constructible_set object; see the type definition in ConstructibleSetTools.

A semi-algebraic set must be encoded by a list of regular_semi_algebraic_system, see the type definition of RealTriangularize.

The command Difference(cs1, cs2, R) returns a constructible set which is the set theoretic difference of two constructible sets. The polynomial ring may have characteristic zero or a prime characteristic.

The command Difference(lrsas1, lrsas2, R) returns a list of regular semi-algebraic systems, encoding the set-theoretic difference of two semi-algebraic sets. The polynomial ring must have characteristic zero.

The output may contain certain redundancy among the defining regular systems.

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:-Difference or RegularChains:-SemiAlgebraicSetTools:-Difference.

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

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔2x^2+3xy+y^2-3x-3y$

$F≔2x^2+3xy+y^2-3x-3y$

$G≔x{y}^2-x$

$G≔x{y}^2-x$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\left[F\,G\right]\,R\,\mathrm{output}=\mathrm{lazard}\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

$\left[\left[x-2\,y+1\right]\,\left[x-1\,y-1\right]\,\left[x+1\,y-1\right]\,\left[x\,y-3\right]\,\left[x-1\,y+1\right]\,\left[x\,y\right]\right]$

$H≔y^3-y$

$H≔y^3-y$

$\mathrm{rcH}≔\mathrm{Chain}\left(\left[H\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rcH}≔\mathrm{regular\_chain}$

$\mathrm{cs1}≔\mathrm{ConstructibleSet}\left(\left[\mathrm{RegularSystem}\left(\mathrm{rcH}\,R\right)\right]\,R\right)$

$\mathrm{cs1}≔\mathrm{constructible\_set}$

$\mathrm{cs2}≔\mathrm{ConstructibleSet}\left(\mathrm{map}\left(\mathrm{RegularSystem}\,\mathrm{dec}\,R\right)\,R\right)$

$\mathrm{cs2}≔\mathrm{constructible\_set}$

$\mathrm{cs3}≔\mathrm{Difference}\left(\mathrm{cs2}\,\mathrm{cs1}\,R\right)$

$\mathrm{cs3}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{cs2}\,R\right)$

$\left[\left[x-2\,y+1\right]\,\left[1\right]\right],\left[\left[x-1\,y-1\right]\,\left[1\right]\right],\left[\left[x+1\,y-1\right]\,\left[1\right]\right],\left[\left[x\,y-3\right]\,\left[1\right]\right],\left[\left[x-1\,y+1\right]\,\left[1\right]\right],\left[\left[x\,y\right]\,\left[1\right]\right]$

$\mathrm{Info}\left(\mathrm{cs3}\,R\right)$

$\left[\left[x\,y-3\right]\,\left[1\right]\right]$

$\mathrm{C1}≔\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;b^2+a^2-c^2\le 0\right]\&colon;$

$\mathrm{C2}≔\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;c{\left(b^2+a^2-c^2\right)}^2<a{b}^2\left(2ac-\left(c^2+a^2-b^2\right)\right)\right]\&colon;$

$\mathrm{S1}≔\left[\mathrm{C1}\&comma;\mathrm{C2}\right]$

$\mathrm{S1}≔\left[\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;a^2+b^2-c^2\le 0\right]\&comma;\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;c{\left(a^2+b^2-c^2\right)}^2<a{b}^2\left(-a^2+2ac+b^2-c^2\right)\right]\right]$

$\mathrm{S2}≔\left[a-c<0\&comma;0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\right]$

$\mathrm{S2}≔\left[a-c<0\&comma;0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\right]$

$R≔\mathrm{PolynomialRing}\left(\left[a\&comma;b\&comma;c\right]\right)\&colon;$

$\mathrm{dec1}≔\mathrm{map}\left(\mathrm{op}\&comma;\mathrm{map}\left(\mathrm{RealTriangularize}\&comma;\mathrm{S1}\&comma;R\right)\right)$

$\mathrm{dec1}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{dec2}≔\mathrm{RealTriangularize}\left(\mathrm{S2}\&comma;R\right)$

$\mathrm{dec2}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Difference}\left(\mathrm{dec1}\&comma;\mathrm{dec2}\&comma;R\right)$

$\mathrm{Difference}\left(\mathrm{dec2}\&comma;\mathrm{dec1}\&comma;R\right)$

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][Difference] command was introduced in Maple 16.

The lrsas1 and lrsas2 parameters were introduced in Maple 16.

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