topMinimalPrimesIP -- compute the minimal primes of maximum dimension using integer programming

Synopsis

Usage:

topMinimalPrimesIP(I)

topMinimalPrimesIP(I, KnownDim => k)

topMinimalPrimesIP(I, IgnorePrimes => P)
Inputs:
- I, a monomial ideal,
Optional inputs:
- KnownDim => an integer, default value -1, the dimension, k, of the ideal
- IgnorePrimes => a list, default value {}, a list of primes to not include in the result. See IgnorePrimes.
Outputs:
- L, a list, all minimal associated primes of dimension $k$

Description

If a KnownDim is not provided, topMinimalPrimesIP will first call {dimensionIP}($I$) to compute the dimension.

The IP for this function is similar to the degreeIP formulation, except that rather than count the number of solutions, SCIP uses a sparse data structure to enumerate all feasible solutions.

The location of input/output files for SCIP solving is printed to the screen by default. To change this, see ScipPrintLevel.

i1 : R = QQ[x,y,z,w,v];

i2 : I = monomialIdeal(y^12, x*y^3, z*w^3, z*v*y^10, z*x^10, v*z^10, w*v^10, y*v*x*z*w);

o2 : MonomialIdeal of R

i3 : ScipPrintLevel = 0;

i4 : minimalPrimes(I)

o4 = {monomialIdeal (y, z, w), monomialIdeal (y, z, v), monomialIdeal (x, y,
     ------------------------------------------------------------------------
     w, v)}

o4 : List

i5 : apply(oo, p -> dim p)

o5 = {2, 2, 1}

o5 : List

i6 : topMinimalPrimesIP(I)

o6 = {monomialIdeal (y, z, v), monomialIdeal (y, z, w)}

o6 : List

Notice that if the dimension of a monomial ideal is $k$, each of the top minimal primes is generated by $n-k$ variables, where $n$ is the number of variables in the polynomial ring.

Caveat

topMinimalPrimesIP does not verify that a provided KnownDim is correct. Providing the wrong dimension will result in an incorrect answer or an error.

Ways to use topMinimalPrimesIP :

topMinimalPrimesIP(MonomialIdeal)

For the programmer

The object topMinimalPrimesIP is a method function with options.