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CheckUnmixed -- check whether ideals encountered are unmixed


If set to false, the program never checks whether the ideal $I$ or any $C_{y,I}$ or $N_{y,I}$ ideals are unmixed. Setting CheckUnmixed=>false will speed up computations since it is not performing a check of this condition but comes at the cost that not all the necessary conditions are checked. Notice that if isGVD(I, CheckUnmixed=>false) returns false, then $I$ is conclusively not geometrically vertex decomposable as there is some other condition that is not met. The default value is true.

If you know that $I$ is unmixed but want to check unmixedness for $C_{y,I}$, $N_{y,I}$, and any later ideals, use IsIdealUnmixed instead.

The following is not unmixed [SM, Example 1.6] and hence not geometrically vertex decomposable. However, if we disable the unmixedness check and skip the Cohen-Macaulay check, isGVD returns true.

i1 : R = QQ[x_1..x_5]

o1 = R

o1 : PolynomialRing
i2 : I = ideal(x_1*x_3, x_1*x_4, x_1*x_5, x_2*x_3, x_2*x_4, x_2*x_5)

o2 = ideal (x x , x x , x x , x x , x x , x x )
             1 3   1 4   1 5   2 3   2 4   2 5

o2 : Ideal of R
i3 : isUnmixed I

o3 = false
i4 : isGVD(I, CheckCM=>"never", CheckUnmixed=>false)

o4 = true


[SM] Hero Saremi and Amir Mafi. Unmixedness and Arithmetic Properties of Matroidal Ideals. Arch. Math. 114 (2020) 299–304.


As in the above example, if you set CheckUnmixed=>false and you do not already know that both $I$ is unmixed and all later $C_{y,I}$ and $N_{y,I}$ ideals are unmixed, then the output of isGVD or any other GVD method cannot definitely conclude that $I$ is geometrically vertex decomposable, as not all of conditions in the definition were checked.

See also

Functions with optional argument named CheckUnmixed :

For the programmer

The object CheckUnmixed is a symbol.