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 HilbertSeries

the Hilbert-Poincare series
 Syntax
 ``` HilbertSeries(M:RING or TAGGED("Quotient")):TAGGED("\$hp.PSeries") ```

 Description
This function computes the Hilbert-Poincare series of M. The input, M, must be homogeneous (with respect to the first row of the weights matrix). In the standard case, i.e. the weights of all indeterminates are 1, the result is simplified so that the power appearing in the denominator is the dimension of M.

The function Poincare is exacly the same as HilbertSeries .

NOTES:

(i) the coefficient ring must be a field.

(ii) these functions produce tagged objects: they cannot safely be (non-)equality to other values.

For more information, see the article: A.M. Bigatti, "Computations of Hilbert-Poincare Series" J. Pure Appl. Algebra, 119/3 (1997), 237--253.

 Example
 ``` Use R ::= QQ[t,x,y,z]; HilbertSeries(R/Ideal(0)); (1) / (1-t)^4 ------------------------------- Q := R/Ideal(t^2,x,y^3); Poincare(Q); (1 + 2t + 2t^2 + t^3) / (1-t) ------------------------------- Poincare(R^2/Module([x^2,y],[z,y])); (2 + t) / (1-t)^3 ------------------------------- Use R ::= QQ[t,x,y,z], Weights([1,2,3,4]); Poincare(R/Ideal(t^2,x,y^3)); --- Non Simplified Pseries --- (1-2t^2 + t^4 - t^9 + 2t^11 - t^13) / ( (1-t) (1-t^2) (1-t^3) (1-t^4) ) ------------------------------- Use R ::= QQ[t,x,y,z], Weights(Mat([[1,2,3,4],[0,0,5,8]])); Poincare(R/Ideal(t^2,x,y^3)); --- Non Simplified Pseries --- ( - t^13x^15 + 2t^11x^15 - t^9x^15 + t^4-2t^2 + 1) / ( (1-t) (1-t^2) (1-t^3x^5) (1-t^4x^8) ) ------------------------------- ```