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SymbolicPowers :: symbolicPower

symbolicPower -- computes the symbolic power of an ideal.

Synopsis

Description

Given an ideal I and an integer n, this method returns the n-th symbolic power of I. Various algorithms are used, in the following order:

1. If I is a homogeneous ideal in a polynomial ring whose height is one less than the dimension of the ring, returns the saturation of I;

2. If I is squarefree monomial ideal, intersects the powers of the associated primes of I

3. If I is monomial ideal, but not squarefree, takes a primary decomposition of I, picks the maximal elements in it, and intersects their powers;

4. If I is prime, computes a primary decomposition of In and intersects the components with radical I.

5. If all else fails, compares the radicals of a primary decomposition of In with the associated primes of I, and intersects the components corresponding to minimal primes.

i1 : B = QQ[x,y,z];
i2 : f = map(QQ[t],B,{t^3,t^4,t^5})

                   3   4   5
o2 = map(QQ[t],B,{t , t , t })

o2 : RingMap QQ[t] <--- B
i3 : I = ker f;

o3 : Ideal of B
i4 : symbolicPower(I,2)

             4       2     2 2   2 3    3       2 2      3   3 2    4     3 
o4 = ideal (y  - 2x*y z + x z , x y  - x y*z - y z  + x*z , x y  - x z - y z
     ------------------------------------------------------------------------
            2   5      3     2       3
     + x*y*z , x  + x*y  - 3x y*z + z )

o4 : Ideal of B

When computing symbolic powers of a quasi-homogeneous ideal, the method runs faster if the ideal is changed to be homegeneous.

i5 : P = ker map(QQ[t],QQ[x,y,z],{t^3,t^4,t^5})

             2         2     2   3
o5 = ideal (y  - x*z, x y - z , x  - y*z)

o5 : Ideal of QQ[x, y, z]
i6 : isHomogeneous P

o6 = false
i7 : time symbolicPower(P,4);
     -- used 0.59352 seconds

o7 : Ideal of QQ[x, y, z]
i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})

             2         3         2     2
o8 = ideal (y  - x*z, x  - y*z, x y - z )

o8 : Ideal of QQ[x, y, z]
i9 : isHomogeneous Q

o9 = true
i10 : time symbolicPower(Q,4);
     -- used 0.0291216 seconds

o10 : Ideal of QQ[x, y, z]

See also

Ways to use symbolicPower :