This checks whether the generators of a subring or the sagbi generators of a computation object form a sagbi basis. After running sagbi on a subring, note that the result of the computation is stored in that subring. If new sagbi generators need to be added to S to make it a sagbi basis then isSAGBI will return false. If isSAGBI is run on a ring for which no sagbi computation has been performed, then isSAGBI will perform a test to see if the generators form a sagbi basis. This test can be skipped by setting Compute to false, and, in that case, isSAGBI will return null.
i1 : R = QQ[x,y,z];
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i2 : S = subring {x+y+z,x*y+x*z+y*z, x*y*z, (x-y)*(x-z)*(y-z)};
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i3 : isSAGBI S
o3 = false
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i4 : sagbi(S, Limit => 5)
o4 = Partial SAGBIBasis Computation Object with 6 generators, Limit = 5.
o4 : SAGBIBasis
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i5 : isSAGBI S
-- 0.000184705 seconds elapsed
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timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
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o5 = false
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i6 : S' = subring {x+y+z,x*y+x*z+y*z, x*y*z};
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i7 : isSAGBI S'
o7 = true
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i8 : sagbi(S', Limit => 5)
o8 = SAGBIBasis Computation Object with 3 generators, Limit = 5.
o8 : SAGBIBasis
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i9 : isSAGBI S'
-- 0.000742862 seconds elapsed
-- 0.00019789 seconds elapsed
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-- 0.000209483 seconds elapsed
-- 0.000274344 seconds elapsed
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
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timing raw subduction
o9 = true
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If isSAGBI is supplied a SAGBIBasis then the generators of the subring can be checked for being a sagbi basis by setting UseSubringGens to true.
A note of caution. Excessive use of isSAGBI may impact performance. The inputs and outputs of isSAGBI are stored and never deleted. Since the inputs and outputs include pointers objects such as SAGBIBasis computation objects, these objects will not be garbage collected.