# The Packing Problem

Given a square-free monomial ideal $I$ of codimension $c$, $I$ is Konig if it contains a regular sequence of monomials of length $c$.

We can test if a given ideal is Konig:

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x*y,z*y,x*z) o2 = ideal (x*y, y*z, x*z) o2 : Ideal of R i3 : isKonig(I) o3 = false

$I$ is said to have the packing property if any ideal obtained from $I$ by setting any number of variables equal to 0 is Konig.

 i4 : isPacked(I) o4 = false

Given an ideal that is not packed, we can determine which variable substitutions lead to ideals that are not Konig.

 i5 : noPackedAllSubs(I) o5 = Only I is not Konig -- all proper substitutions are Konig.

We can obtained just one substitution leading to an ideal that is not Konig, or all such substitutions:

 i6 : R = QQ[a,b,c,d,A,B,C,D] o6 = R o6 : PolynomialRing i7 : J = ideal"ABCD,cdAB,abcD,bcABD,abcBD,abcdA,abcAD,abdAC,acdBC,acBCD,abcdC,bcdAC,bcACD,bcdAD,acdBD,adBCD,acABD,bdABC,adABC" o7 = ideal (A*B*C*D, c*d*A*B, a*b*c*D, b*c*A*B*D, a*b*c*B*D, a*b*c*d*A, ------------------------------------------------------------------------ a*b*c*A*D, a*b*d*A*C, a*c*d*B*C, a*c*B*C*D, a*b*c*d*C, b*c*d*A*C, ------------------------------------------------------------------------ b*c*A*C*D, b*c*d*A*D, a*c*d*B*D, a*d*B*C*D, a*c*A*B*D, b*d*A*B*C, ------------------------------------------------------------------------ a*d*A*B*C) o7 : Ideal of R i8 : isPacked(J) o8 = false i9 : noPackedSub(J) o9 = The ideal itself is not Konig!

These can easily be tested:

 i10 : L = substitute(J,matrix{{1,0,c,d,A,1,C,D}}) o10 = ideal (A*C*D, c*d*A, 0, 0, 0, 0, 0, 0, c*d*C, c*C*D, 0, 0, 0, 0, c*d*D, ----------------------------------------------------------------------- d*C*D, c*A*D, 0, d*A*C) o10 : Ideal of R i11 : isKonig(L) o11 = false