This method computes the Hilbert series of a graded noncommutative ring. If the ring is defined over a field (and potentially not standard graded), then a basis is computed and the generating function of the degrees of that basis is returned. The degree to which one computes the Hilbert series is controlled with the Order option. The output is returned as either an expression (if a rational representation can be found using toRationalFunction) or as an element of the degreesRing of the input.
i1 : A = QQ<|x,y,z|> o1 = A o1 : FreeAlgebra |
i2 : ncHilbertSeries(A,Order=>10) 1 o2 = ------ 1 - 3T o2 : Expression of class Divide |
i3 : A = QQ<|x,y,z,Degrees=>{1,2,3}|> o3 = A o3 : FreeAlgebra |
i4 : ncHilbertSeries(A,Order=>10) 1 o4 = ----------------- 2 3 1 - (T + T + T ) o4 : Expression of class Divide |
i5 : B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z}) o5 = B o5 : FreeAlgebraQuotient |
i6 : ncHilbertSeries(B,Order=>10) 1 o6 = -------- 3 (1 - T) o6 : Expression of class Divide |
The object ncHilbertSeries is a method function with options.