Hilbert Series

The following functions compute the Hilbert series information of graded or (homogeneous) modules. This depends on the column weights, just as in graded polynomial rings.

HilbertSeries(M): ModMPol -> FldFunElt

Given a graded \(R\)-module \(M\), return the Hilbert series \(H_M(t)\) of \(M\) (as an element of the univariate function field over the ring of integers). The \(i\)-th coefficient of the series gives the vector-space dimension of the degree-\(i\) graded piece of \(M\). The algorithm implemented is that given in [Bayer and Stillman, 1992]. Note that if \(I\) is an ideal of the ring \(R\), then the corresponding function for ideals HilbertSeries applied to \(I\) gives the Hilbert series of the affine algebra (quotient) \(R/I\), so this is equivalent to HilbertSeries(QuotientModule(I)).

HilbertSeries(M, p): ModMPol, RngIntElt -> RngSerLaurElt

Given a graded \(R\)-module \(M\), return the Hilbert series \(H_M(t)\) of \(M\) (as a Laurent series to precision \(p\)). (A Laurent series is required in general, since negative powers may occur when there are negative values in the grading of \(M\).)

HilbertDenominator(M): ModMPol -> RngUPol

Given a graded \(R\)-module \(M\), return the unreduced Hilbert denominator \(D\) of the Hilbert series \(H_M(t)\) of \(M\) (as a univariate polynomial over the ring of integers). The denominator \(D\) equals HilbertDenominator(R) which is simply \(\prod_{i=1}^n (1-t^{w_i}),\) where \(n\) is the rank of \(R\) and \(w_i\) is the weight of the \(i\)-th variable (1 by default).

HilbertNumerator(M): ModMPol -> RngUPolElt, RngIntElt

Given a graded \(R\)-module \(M\), return the unreduced Hilbert numerator \(N\) of the Hilbert series \(H_M(t)\) of \(M\) (as a univariate polynomial over the ring of integers) and a valuation shift \(s\). The numerator \(N\) equals \(D\times t^s \times H_M(t)\), where \(D\) is the unreduced Hilbert denominator above. Computing with the unreduced numerator is often more convenient. Note that \(s\) will only be non-zero when \(M\) has negative weights in its grading.

HilbertPolynomial(I): ModMPol -> RngUPolElt, RngIntElt

Given a graded \(R\)-module \(M\), return the Hilbert polynomial \(H(d)\) of \(M\) as an element of the univariate polynomial ring \({\mathbb{Q}}[d]\), together with the index of regularity of \(M\) (the minimal integer \(k \geq 0\) such that \(H(d)\) agrees with the Hilbert function of \(M\) at \(d\) for all \(d \geq k\)).

Example: Hilbert (ex-ffdbad)

We apply the Hilbert series functions to a simple quotient module.

> R<x,y,z> := PolynomialRing(RationalField(), 3);
> F := GradedModule(R, 3);
> M := quo<F | [x,0,0], [0,y^2,0]>;
> M;
Graded Module R^3/<relations>
Relations:
[  x,   0,   0],
[  0, y^2,   0]
> HilbertSeries(M);
(t^2 + t - 3)/(t^3 - 3*t^2 + 3*t - 1)
> HilbertSeries(M, 10);
3 + 8*s + 14*s^2 + 21*s^3 + 29*s^4 + 38*s^5 + 48*s^6 + 59*s^7 + 71*s^8 + 84*s^9
    + O(s^10)
> HilbertNumerator(M);
-x^2 - x + 3
0
> HilbertDenominator(M);
-x^3 + 3*x^2 - 3*x + 1
> HilbertPolynomial(M);
1/2*x^2 + 9/2*x + 3
0
> [Evaluate(HilbertPolynomial(F), i): i in [0..10]];
[ 3, 9, 18, 30, 45, 63, 84, 108, 135, 165, 198 ]

Run in calculatorCopy input

If the module has negative weights, then denominator may include extra powers of \(t\), so the shift for the numerator will be non-zero.

> F := GradedModule(R, [-1]);
> F;
Free Graded Module R^1 with grading [-1]
> HilbertSeries(F);
-1/(t^4 - 3*t^3 + 3*t^2 - t)
> HilbertSeries(F, 10);
s^-1 + 3 + 6*s + 10*s^2 + 15*s^3 + 21*s^4 + 28*s^5 + 36*s^6 + 45*s^7 + O(s^8)
> HilbertNumerator(F);
1
1
> HilbertDenominator(F);
-x^3 + 3*x^2 - 3*x + 1
> HilbertPolynomial(F);
1/2*x^2 + 5/2*x + 3
-1
> [Evaluate(HilbertPolynomial(F), i): i in [-1..10]];
[ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78 ]

Run in calculatorCopy input