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# isSAGBI -- Check if the generators are a subalgebra basis

## Synopsis

• Usage:
result = isSAGBI S
result = isSAGBI SB
result = isSAGBI M
result = isSAGBI L
• Inputs:
• S, ,
• SB, ,
• M, ,
• L, a list,
• Outputs:
• result, , if true, then the generators or subalgebra generators form a subalgebra basis.

## Description

Checks whether the generators of or the subalgebra generators of form a subalgebra basis. If isSAGBI is supplied or a list A, then the command isSAGBI A is equivalent to isSAGBI subring A. For further details of their respective uses see isSAGBI(Subring) and isSAGBI(SAGBIBasis).

The result of isSagbi is stored in the cache of or the sagbiStatus of . If isSagbi is called again, then the result is looked up, unless the option Recompute is set to true.

 i1 : R = QQ[x,y,z]; i2 : S = subring {x+y+z,x*y+x*z+y*z, x*y*z, (x-y)*(x-z)*(y-z)}; i3 : isSAGBI S o3 = false i4 : isSAGBI sagbi(S, Limit => 5) o4 = false i5 : S' = subring {x+y,x*y,x*y^2,x*y^4-y}; i6 : isSAGBI S' o6 = false i7 : isSAGBI sagbi(S', Limit => 10) o7 = true

If isSAGBI is supplied , then the generators of its subring can be checked for being a subalgebra basis by setting UseSubringGens to true. If isSAGBI is supplied , then the generators of a partial subalgebra basis can be checked for being a subalgebra basis by setting UseSubringGens to false.

Optional inputs for specific uses (see isSAGBI(Subring) and isSAGBI(SAGBIBasis)) include the following: Compute, Recompute, Subduction strategies, SubductionMethod, PrintLevel, RenewOptions, UseSubringGens, and ModifySAGBIBasis.

## For the programmer

The object isSAGBI is .