A way of computing the Hilbert series
Let S = K[x1, x2, . . . , xn] be a standard graded K-algebra for any field K. Without using any heavy tools of commutative algebra we compute the Hilbert series of graded S-module S/I, where I is a monomial ideal.
Збережено в:
| Дата: | 2018 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2018
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188346 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A way of computing the Hilbert series / A. Haider // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 35-38. — Бібліогр.: 4 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-188346 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1883462023-02-24T01:27:20Z A way of computing the Hilbert series Haider, A. Let S = K[x1, x2, . . . , xn] be a standard graded K-algebra for any field K. Without using any heavy tools of commutative algebra we compute the Hilbert series of graded S-module S/I, where I is a monomial ideal. 2018 Article A way of computing the Hilbert series / A. Haider // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 35-38. — Бібліогр.: 4 назв. — англ. 1726-3255 2010 MSC: Primary 13P10; Secondary 13F20, 68R05, 05E40. http://dspace.nbuv.gov.ua/handle/123456789/188346 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Let S = K[x1, x2, . . . , xn] be a standard graded K-algebra for any field K. Without using any heavy tools of commutative algebra we compute the Hilbert series of graded S-module S/I, where I is a monomial ideal. |
| format |
Article |
| author |
Haider, A. |
| spellingShingle |
Haider, A. A way of computing the Hilbert series Algebra and Discrete Mathematics |
| author_facet |
Haider, A. |
| author_sort |
Haider, A. |
| title |
A way of computing the Hilbert series |
| title_short |
A way of computing the Hilbert series |
| title_full |
A way of computing the Hilbert series |
| title_fullStr |
A way of computing the Hilbert series |
| title_full_unstemmed |
A way of computing the Hilbert series |
| title_sort |
way of computing the hilbert series |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188346 |
| citation_txt |
A way of computing the Hilbert series / A. Haider // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 35-38. — Бібліогр.: 4 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT haidera awayofcomputingthehilbertseries AT haidera wayofcomputingthehilbertseries |
| first_indexed |
2025-07-16T10:22:02Z |
| last_indexed |
2025-07-16T10:22:02Z |
| _version_ |
1837798599848624128 |
| fulltext |
“adm-n1” — 2018/4/2 — 12:46 — page 35 — #37
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 1, pp. 35–38
c© Journal “Algebra and Discrete Mathematics”
A way of computing the Hilbert series
Azeem Haider
Communicated by I. P. Shestakov
Abstract. Let S = K[x1, x2, . . . , xn] be a standard graded
K-algebra for any field K. Without using any heavy tools of com-
mutative algebra we compute the Hilbert series of graded S-module
S/I, where I is a monomial ideal.
Let S = K[x1, x2, . . . , xn] be a standard graded K-algebra for a field K.
We present a way of computing the Hilbert series of a graded S-module
S/I, when I is a monomial ideal of S. There are some known ways of
computing the Hilbert series (for example [3]). Unlike any other method
we compute Hilbert series of S/I by skipping heavy tools of commutative
algebra.
If I ⊂ S is a monomial ideal which is minimally generated by mono-
mials {u1, u2, . . . , us}, Sd/Id is the dth graded component of S/I and
H(S/I, d) = dimK Sd/Id is the dimension of Sd/Id as a K linear space,
called the Hilbert function of S/I. The series
HS/I(t) =
∞
∑
d=0
H(S/I, d)td
is called the Hilbert Series of S/I. To know more about Hilbert series of
general graded modules and related results see [1], [2] or [4].
2010 MSC: Primary 13P10; Secondary 13F20, 68R05, 05E40.
Key words and phrases: monomial ideal, Hilbert series.
“adm-n1” — 2018/4/2 — 12:46 — page 36 — #38
36 A way of computing the Hilbert series
For any monomials u =
n
∏
i=0
xaii and v =
n
∏
i=0
xbii in S, we define the
intersecting multiplication as
u ∗ v =
n
∏
i=0
x
max{ai,bi}
i . (1)
Considering this sort of multiplication allowing us to associate a
monomial ideal I = (u1, u2, . . . , us) ⊂ S to a unique polynomial called
intersecting polynomial in the following way
PS/I = (1− u1) ∗ (1− u2) ∗ . . . ∗ (1− us).
Simplifying we get
PS/I = 1−
s
∑
i=1
(ui)+
s
∑
16i1<i2
(ui1 ∗ui2)− . . .+(−1)s(ui1 ∗ui2 ∗ . . .∗uis). (2)
In the polynomial PS/I we have 2s number of monomial terms with half
positive and half negative coefficient. For simplicity, we denote monomials
with coefficient +1 by v1, v2, . . . , v2s−1 and monomials with coefficient −1
by w1, w2, . . . , w2s−1 . Using these notations we can write (2) as
PS/I =
2s−1
∑
i=1
(vi − wi). (3)
Note that if an ideal J = (I, u), for monomial u /∈ I, then PS/J =
PS/I − u ∗ PS/I .
Once we obtained the intersecting polynomial PS/I , we can write
Hilbert series of S/I as described in the following theorem.
Theorem 1. If PS/I =
∑2s−1
i=1 (vi − wi) is the intersecting polynomial of
S/I for monomial ideal I ⊂ S, then
HS/I(t) =
2s−1
∑
i=1
(
tdeg(vi) − tdeg(wi)
)
(1− t)n
.
Proof. If I = (u1, u2, . . . , us) ⊂ S is a monomial ideal, then by inclusion
exclusion principal we can see that the dimension of dth graded component
of I is
H(I, d) =
s
∑
i=1
|ui|d−
s
∑
16i1<i2
|ui1 ∗ui2 |d− . . .+(−1)s+1|ui1 ∗ui2 ∗ . . .∗uis |d,
“adm-n1” — 2018/4/2 — 12:46 — page 37 — #39
A. Haider 37
where |u|d =
(
n− 1 + d− deg(u)
n− 1
)
is the dimension of dth graded com-
ponent of an ideal generated by a monomial u ∈ S.
Now H(S/I, d) = H(S, d)−H(I, d), for all d and H(S, d) = |1|d, hence
H(S/I, d) = |1|d −
s
∑
i=1
|ui|d
+
s
∑
16i1<i2
|ui1 ∗ ui2 |d − . . .+ (−1)s|ui1 ∗ ui2 ∗ . . . ∗ uis |d.
(4)
If we replace each monomial term in the intersecting polynomial defined
in (2) by the corresponding dth graded component of ideal generated by
that monomial, then we obtain H(S/I, d) as in (4). Hence we can write
Hilbert function H(S/I, d) in terms of (3) as
H(S/I, d) =
2s−1
∑
i=1
(|vi|d − |wi|d)
=
2s−1
∑
i=1
((
n− 1 + d− deg(vi)
n− 1
)
−
(
n− 1 + d− deg(wi)
n− 1
))
and the corresponding Hilbert series is
HS/I(t) =
∞
∑
d=0
2s−1
∑
i=1
(|vi|d − |wi|d)
=
2s−1
∑
i=1
(
tdeg(vi) − tdeg(wi)
)
(1− t)n
.
We give an example to illustrate our method.
Example. If I = (x21x3, x1x2x
2
3, x
2
2x
3
3) ⊂ S = K[x1, x2, x3], then the
corresponding intersecting polynomial is
PS/I = (1− x21x3) ∗ (1− x1x2x
2
3) ∗ (1− x22x
3
3)
= 1− x21x3 − x1x2x
2
3 − x22x
3
3 + x21x2x
2
3 + x21x
2
2x
3
3 + x1x
2
2x
3
3 − x21x
2
2x
3
3.
The monomials with coefficient +1 and −1 in PS/I are
v1 = 1, v2 = x21x2x
2
3, v3 = x21x
2
2x
3
3, v4 = x1x
2
2x
3
3
“adm-n1” — 2018/4/2 — 12:46 — page 38 — #40
38 A way of computing the Hilbert series
and
w1 = x21x3, w2 = x1x2x
2
3, w3 = x22x
3
3, w4 = x21x
2
2x
3
3,
respectively.
Now by using Theorem 1 we can write Hilbert series of S/I, that is;
HS/I(t) =
t0 − t3 + t5 − t4 + t7 − t5 + t6 − t7
(1− t)3
=
1 + t+ t2 − t4 − t5
(1− t)2
.
References
[1] A. Ya. Belov, V. V. Borisenko, V. N. Latyshev, Monomial algebras, Journal of
Mathematical Sciences, Vol. 87, Issue 3, November 1997, pp. 3463-3575.
[2] W. Bruns and J. Herzog, Cohen–Macaulay rings, Revised Edition, Cambridge
University Press, 1998.
[3] M. Stillman and D. Bayer, Computations of Hilbert functions, J. Symbolic Com-
putations, 14, 1992, pp. 31-50.
[4] V.A. Ufnarovskij, Combinatorial and asymptotic methods in algebra, (Itogi Nauki
Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 57, pp. 5-177, 1990).
Translation: Algebra VI. Encycl. Math. Sci. 57, 1995, pp. 1-196.
Contact information
A. Haider Department of Mathematics, Jazan University,
Jazan, Saudi Arabia
E-Mail(s): aahaider@jazanu.edu.sa
Received by the editors: 09.03.2016
and in final form 23.03.2017.
|