symbPowerPrimePosChar

Synopsis

Usage:

symbPowerPrimePosChar(I,n)
Inputs:
- I, an ideal,
- n, an integer,
Outputs:
- an ideal, the n-th symbolic power of I

Description

Given a prime ideal $I$ in a polynomial ring over a field of positive characteristic, and an integer $n$, this method returns the $n$-th symbolic power of $I$. To compute $I^{(a)}$, find the largest value $k$ with $q = p^k \leq a$. Then $I^{(a)} = (I^{[q]} : I^{a-q+1})$.

i1 : B = ZZ/7[x,y,z];

i2 : f = map(ZZ/7[t],B,{t^3,t^4,t^5})

          ZZ          3   4   5
o2 = map (--[t], B, {t , t , t })
           7

             ZZ
o2 : RingMap --[t] <-- B
              7

i3 : I = ker f;

o3 : Ideal of B

i4 : symbPowerPrimePosChar(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

Caveat

The ideal must be prime.

Ways to use symbPowerPrimePosChar :

symbPowerPrimePosChar(Ideal,ZZ)

For the programmer