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Poincare, HilbertSeries

Syntax

Poincare(M:RING or TAGGED("Quotient")):TAGGED("$hp.PSeries")
HilbertSeries(M:RING or TAGGED("Quotient")):TAGGED("$hp.PSeries")
PoincareShifts(M: Module, ShiftsList: LIST):TAGGED("$hp.PSeries")
PoincareShifts(M: TAGGED("Quotient"), ShiftsList: LIST)
                                           :TAGGED("$hp.PSeries")
    

Summary

the Poincare series

Description

These functions all compute the Hilbert-Poincare series of M. The first two functions perform the same operations. The input, M, must be homogeneous (with respect to the first row of the weights matrix). In the standard case, where the weights of all indeterminates are 1, the result is simplified so that the power appearing in the denominator is the dimension of M.

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 ::= Q[t,x,y,z];
Poincare(R);
(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 ::= Q[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 ::= Q[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) )
-------------------------------
Use P ::= Q[x,y,z];
M := Module([x,y^3], [x-z,0]);
PoincareShifts(M, [2,0]);     -- Poincare series of a shifted module
(2x^3) / (1-x)^3
-------------------------------
PoincareShifts(P^2/M, [3,1]); -- Poincare series of a shifted quotient module
(x + x^2 + 2x^3) / (1-x)^2
-------------------------------
        
    

See Also