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SegreClasses :: isComponentContained

isComponentContained -- This method tests containment of (irreducible) varieties; more generally it tests if a top-dimensional irreducible component of the scheme associated an ideal is contained in the scheme associated to another ideal

Synopsis

Description

For a subschemes X of an irreducible subscheme Y of ℙn1x...xℙnm this command tests whether or not a top-dimensional irreducible (and reduced) component of X is contained in Y

i1 : R = makeProductRing({2,2,2})

o1 = R

o1 : PolynomialRing
i2 : x=(gens R)_{0..2}

o2 = {a, b, c}

o2 : List
i3 : y=(gens R)_{3..5}

o3 = {d, e, f}

o3 : List
i4 : z=(gens R)_{6..8}

o4 = {g, h, i}

o4 : List
i5 : m1=matrix{{x_0,x_1,5*x_2},y_{0..2},{2*z_0,7*z_1,25*z_2}}

o5 = | a  b  5c  |
     | d  e  f   |
     | 2g 7h 25i |

             3       3
o5 : Matrix R  <--- R
i6 : m2=matrix{{9*z_0,4*z_1,3*z_2},y_{0..2},x_{0..2}}

o6 = | 9g 4h 3i |
     | d  e  f  |
     | a  b  c  |

             3       3
o6 : Matrix R  <--- R
i7 : W=minors(3,m1)+minors(3,m2);

o7 : Ideal of R
i8 : f=random({1,1,1},R);
i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f);

o9 : Ideal of R
i10 : X=((W)*ideal(y)+ideal(f));

o10 : Ideal of R
i11 : time isComponentContained(X,Y)
     -- used 2.13869 seconds

o11 = true
i12 : print "we could confirm this with the computation:"
we could confirm this with the computation:
i13 : B=ideal(x)*ideal(y)*ideal(z)

o13 = ideal (a*d*g, a*d*h, a*d*i, a*e*g, a*e*h, a*e*i, a*f*g, a*f*h, a*f*i,
      -----------------------------------------------------------------------
      b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g,
      -----------------------------------------------------------------------
      c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i)

o13 : Ideal of R
i14 : time isSubset(saturate(Y,B),saturate(X,B))
     -- used 11.8756 seconds

o14 = true

Ways to use isComponentContained :