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ncHilbertSeries -- Computes the Hilbert series of a noncommutative ring

Description

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)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

         1
o6 = --------
            3
     (1 - T)

o6 : Expression of class Divide

Ways to use ncHilbertSeries:

  • ncHilbertSeries(FreeAlgebra)
  • ncHilbertSeries(FreeAlgebraQuotient)

For the programmer

The object ncHilbertSeries is a method function with options.

The source of this document is in AssociativeAlgebras/doc.m2:1075:0.