# CoCoA:HilbertSeries

## HilbertSeries

the Hilbert-Poincare series

### 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 <ttref>Poincare</ttref> is exacly the same as <ttref>HilbertSeries</ttref>.

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];
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 ::= 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) )
-------------------------------
```

### Syntax

```HilbertSeries(M:RING or TAGGED(<quotes>Quotient</quotes>)):TAGGED(<quotes>\$hp.PSeries</quotes>)
```

```   <type>groebner</type>
<type>groebner-basic</type>
<type>hilbert</type>
<type>quotient</type>
<type>ring</type>
```