|
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
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.
Given a graded R-module M, return the Hilbert series HM(t) of M (as a
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 [BS92].
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)).
Given a graded R-module M, return the Hilbert series HM(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.)
Given a graded R-module M, return the unreduced Hilbert
denominator D of the Hilbert series HM(t) of M (as a univariate
polynomial over the ring of integers).
The denominator D equals
HilbertDenominator(R) which is simply ∏i=1n
(1 - twi), where n is the rank of R and wi is the weight of
the i-th variable (1 by default).
Given a graded R-module M, return the unreduced Hilbert numerator N
of the Hilbert series HM(t) of M (as a univariate polynomial over
the ring of integers) and a valuation shift s.
The numerator N equals D x ts x HM(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.
Given a graded R-module M, return the Hilbert polynomial H(d) of M
as an element of the univariate polynomial ring Q[d], together with
the index of regularity of M (the minimal integer k ≥0 such that
H(d) agrees with the Hilbert function of M at d for all d ≥k).
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 ]
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 ]
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|
