CoCoA:Intersection

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 <command>
   <title>Intersection</title>
   <short_description>intersect lists, ideals, or modules</short_description>
   <description>

The function Intersection returns the intersection of E_1,...,E_n. In the case where the E_i's are lists, it returns the elements common to all of the lists. <par/> The coefficient ring must be a field. <par/> NOTE: In order to compute the intersection of inhomogeneous ideals, it may be faster to use the function HIntersection. To compute the intersection of ideals corresponding to zero-dimensional schemes, see the commands <ttref>GBM</ttref> and <ttref>HGBM</ttref>.

<example>

 Use R ::= Q[x,y,z];
 Points := [[0,0],[1,0],[0,1],[1,1]]; -- a list of points in the plane
 I := Ideal(x,y); -- the ideal for the first point
 Foreach P In Points Do
   I := Intersection(I,Ideal(x-P[1]z,y-P[2]z));
 EndForeach;
 I;  -- the ideal for (the projective closure of) Points

Ideal(y^2 - yz, x^2 - xz)


 Intersection([<quotes>a</quotes>,<quotes>b</quotes>,<quotes>c</quotes>],[<quotes>b</quotes>,<quotes>c</quotes>,<quotes>d</quotes>]);

[<quotes>b</quotes>, <quotes>c</quotes>]


 It = Intersection(Ideal(x,y),Ideal(y^2,z));

TRUE


</example>

   </description>
   <syntax>

Intersection(E_1:LIST,....,E_n:LIST):LIST Intersection(E_1:IDEAL,...,E_n:IDEAL):IDEAL Intersection(E_1:MODULE,....,E_n:MODULE):MODULE </syntax>

   <see>GBM</see>
   <see>HGBM</see>
   <see>IntersectionList</see>
   <see>HIntersection</see>
   <see>HIntersectionList</see>
   <see>GBasis5, and more</see>
   <type>groebner</type>
   <type>groebner-basic</type>
   <type>ideal</type>
   <type>list</type>
   <type>module</type>
   <type>cocoaserver</type>
   <key>intersection</key>
   <key>intersectionlist</key>
 </command>