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# Subduction computation limit -- Bound for subalgebra basis computation

The value of Limit is used internally within the sagbi function to stop the computation before a subalgebra basis has been found. Since subalgebra bases may be infinite, such a barrier is needed to prevent endless loops.

The Limit is an upper bound on the degrees of the $S$-pairs considered by the algorithm in the presentationRing. In this ring, the degrees are assigned according to the degrees of the leading terms of the polynomials that the variables represent.

Since the sagbiDegree does not monotonically increase during the computation, there is no guarantee that a computed partial subalgebra basis is complete up to Limit, even when the computation terminates at this bound.

 i1 : R = QQ[x,y]; i2 : S = subring{x+y,x*y,x*y^2,x*y^5+y}; i3 : T = sagbiBasis S; i4 : sagbiDegree T o4 = -1 i5 : T5 = sagbi(S,Limit=>5) o5 = Partial SAGBIBasis Computation Object with 5 generators, Limit = 5. o5 : SAGBIBasis i6 : sagbiDegree T5 o6 = 6 i7 : gens T5 o7 = | x+y xy xy2 xy3 xy4 | 1 5 o7 : Matrix R <-- R i8 : T20 = sagbi(S,Limit=>20) o8 = SAGBIBasis Computation Object with 6 generators, Limit = 20. o8 : SAGBIBasis i9 : sagbiDegree T20 o9 = 7 i10 : gens T20 o10 = | x+y xy xy2 xy3 xy4 y | 1 6 o10 : Matrix R <-- R