# dimensionIP -- compute the dimension of a monomial ideal using integer programming

## Synopsis

• Usage:
k = dimensionIP(I)
• Inputs:
• I, ,
• Outputs:
• k, an integer, the dimension of $I$. That is, $k$ is the maximum dimension of a coordinate subspace in the variety of $I$.

## Description

This function calls codimensionIP and then returns $n$-codimensionIP($I$), where $n$ is the number of variables in the polynomial ring where $I$ is defined. The integer programming input and output files created will therefore be named "codim.zpl", "codim.errors", etc. as with codimensionIP.

 i1 : R = QQ[x,y,z,w,v]; i2 : I = monomialIdeal(x*y*w, x*z*v, y*x, y*z*v); o2 : MonomialIdeal of R i3 : dimensionIP(I) Codim files saved in directory: /tmp/M2-9918-0/0 o3 = 3

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

 i4 : ScipPrintLevel = 0; i5 : J = monomialIdeal(x*y^3*z^7, y^4*w*v, z^2*v^8, x*w^3*v^3, y^10, z^10) 10 3 7 10 4 3 3 2 8 o5 = monomialIdeal (y , x*y z , z , y w*v, x*w v , z v ) o5 : MonomialIdeal of R i6 : dimensionIP(J) o6 = 2

The dimension of a monomial ideal is equal to the dimension of its radical. Therefore, when looking at the IP formulation written to the temporary file "codim.zpl", you will see that exponents are ignored.