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SymbolicPowers :: The Containment Problem

The Containment Problem

Given an ideal $I$, we can determine if $I^{(m)} \subseteq I^n$. For example, here is an ideal that fails the containment $I^{(3)} \subseteq I^2$:

i1 : B = ZZ/101[x,y,z];
i2 : I = ideal(x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3));

o2 : Ideal of B
i3 : isSymbPowerContainedinPower(I,3,2)

o3 = false

We can also determine the smallest symbolic power contained in a given power.

In our example, $I^{(4)}$ is the smallest symbolic power contained in $I^2$:

i4 : containmentProblem(I,2)

o4 = 4

We can ask the same question backwards: what is the largest power of I that contains $I^{(4)}$?

i5 : containmentProblem(I,4,InSymbolic => true)

o5 = 2