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

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 in RealTriangularize.

The command Intersection(cs1, cs2, R) returns the intersection of two constructible sets.  The polynomial ring may have characteristic zero or a prime characteristic.

The command Intersection(lrsas1, lrsas2, R) returns the intersection of two semi-algebraic sets, encoded by list of regular_semi_algebraic_system. The polynomial ring must have characteristic zero.

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:-Intersection or RegularChains:-SemiAlgebraicSetTools:-Intersection.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$p≔\left(5t+5\right)x-y-\left(10t+7\right)$

$p≔\left(5t+5\right)x-y-10t-7$

$q≔\left(5t-5\right)x-\left(t+2\right)y+\left(-7t+11\right)$

$q≔\left(5t-5\right)x-\left(t+2\right)y-7t+11$

$\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(\left[p\,q\right]\,\left[x-t\right]\,R\right)$

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

$\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[p\,q\right]\,\left[x+t\right]\,R\right)$

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

$\mathrm{cs}≔\mathrm{Intersection}\left(\mathrm{cs1}\,\mathrm{cs2}\,R\right)$

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

$\mathrm{cs3}≔\mathrm{GeneralConstruct}\left(\left[p\,q\right]\,\left[x+t\,x-t\right]\,R\right)$

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

$\mathrm{IsContained}\left(\mathrm{cs3}\,\mathrm{cs}\,R\right)\mathbf{and}\mathrm{IsContained}\left(\mathrm{cs}\,\mathrm{cs3}\,R\right)$

$\mathrm{true}$

$\mathrm{lrsas1}≔\mathrm{RealTriangularize}\left(\left[p\,q\right]\,\left[\right]\,\left[\right]\,\left[x-t\right]\,R\right)$

$\mathrm{lrsas1}≔\left[\mathrm{regular\_semi\_algebraic\_system}\,\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{lrsas2}≔\mathrm{RealTriangularize}\left(\left[p\,q\right]\,\left[\right]\,\left[\right]\,\left[x+t\right]\,R\right)$

$\mathrm{lrsas2}≔\left[\mathrm{regular\_semi\_algebraic\_system}\,\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{lrsas}≔\mathrm{Intersection}\left(\mathrm{lrsas1}\,\mathrm{lrsas2}\,R\right)$

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

$\mathrm{lrsas12}≔\mathrm{RealTriangularize}\left(\left[p\,q\right]\,\left[\right]\,\left[\right]\,\left[x+t\,x-t\right]\,R\right)$

$\mathrm{lrsas12}≔\left[\mathrm{regular\_semi\_algebraic\_system}\,\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Difference}\left(\mathrm{lrsas}\,\mathrm{lrsas12}\,R\right)$

$\mathrm{Difference}\left(\mathrm{lrsas12}\,\mathrm{lrsas}\,R\right)$

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][Intersection] command was introduced in Maple 16.

The lrsas1 parameter was introduced in Maple 16.

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