This method computes the ideal in $R$ of homogeneous polynomial relations on the joint probabilities of random variables represented by the vertices of $G$.
Here is a small example that compute the vanishing ideal on the joint probabilities of two independent binary random variables. In this case, this ideal equals the ideal obtained using conditionalIndependenceIdeal.
i1 : G = digraph {{1,{}}, {2,{}}}
o1 = Digraph{1 => {}}
2 => {}
o1 : Digraph
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i2 : R = markovRing (2,2) o2 = R o2 : PolynomialRing |
i3 : discreteVanishingIdeal (R,G)
o3 = ideal(p p - p p )
1,2 2,1 1,1 2,2
o3 : Ideal of R
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i4 : conditionalIndependenceIdeal(R, localMarkov G)
o4 = ideal(- p p + p p )
1,2 2,1 1,1 2,2
o4 : Ideal of R
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Here is an example for a graph on four vertices. The random variables a,b,c and d have 2,3,4, and 2 states, respectively.
i5 : G = digraph {{a,{b,c}}, {b,{c,d}}, {c,{}}, {d,{}}}
o5 = Digraph{a => {c, b}}
b => {c, d}
c => {}
d => {}
o5 : Digraph
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i6 : R = markovRing (2,3,4,2) o6 = R o6 : PolynomialRing |
i7 : I = discreteVanishingIdeal (R,G); o7 : Ideal of R |
The vanishing ideal is generated by 84 quadrics, which we don't display.
i8 : betti I
0 1
o8 = total: 1 84
0: 1 .
1: . 84
o8 : BettiTally
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The object discreteVanishingIdeal is a method function.