### Università degli Studi di Pisa

### Facoltà di Scienze Matematiche, Fisiche e Naturali

### Corso di Laurea Magistrale in Matematica

## On the Hilbert quasi-polynomials of

## non-standard graded rings

### Candidato:

### Carla Mascia

### Relatore:

### Dott. Massimo Caboara

### Controrelatore:

### Prof.ssa Patrizia Gianni

## Contents

Introduction v

1 Basic knowledge 1

1.1 Graded rings and graded modules . . . 1

1.2 Polynomial rings graded by matrices . . . 3

1.3 Hilbert function and Hilbert-Poincaré series . . . 5

1.4 Hilbert-Serre theorem and Hilbert polynomial . . . 9

2 Hilbert quasi-polynomial 15 2.1 Existence of the Hilbert quasi-polynomial . . . 15

2.2 Degree and leading coefficient of the Hilbert quasi-polynomial 20 2.3 Proprieties of the coefficients of the Hilbert quasi-polynomial 28 2.4 Formulas for the other coefficients of the Hilbert quasi-polynomial . . . 32

3 Algorithm for computing Hilbert quasi-polynomials 45 3.1 Algorithm implementation . . . 45

3.2 Numerical experiments . . . 50

4 Multigraded Hilbert-Poincaré series 55 4.1 Simplicial complexes and corners . . . 55

4.2 The Slice Algorithm . . . 56

4.3 Corner-Euler Algorithm . . . 59

4.4 Computing Hilbert-Poincaré series . . . 61

Conclusions 65

A System of linear Diophantine equations and Smith normal

form 67

B Code of the algorithm for computing Hilbert

quasi-polynomials 71

Bibliography 79

## Introduction

The Hilbert function, its generating function and the Hilbert polynomial of a graded R-module M have been extensively studied since the famous pa-per of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry, e.g. they are an efficient way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.

The main aim of this thesis is to generalize the well-know theory of the Hilbert polynomial for standard graded rings to the general case of rings graded by any vector in Nk

+. The Hilbert function of a non-standard graded

ring is of quasi-polynomial type. We investigate the structure of Hilbert quasi-polynomials, such as degree, leading coefficient and proprieties of their coefficients, and at the same time we present an algorithm to calculate it.

The Hilbert quasi-polynomials have application in several fields, for ex-ample Bruns and Ichim in [Bruns, 2007] have used Hilbert quasi-polynomials to give a purely algebraic proof of an old combinatorial result due to Ehrhart, McMullen and Stanley.

The computation of Hilbert-Poincaré series has also received a lot of at-tention. We consider the problem to get only a part of the Hilbert-Poincaré series, without computing it entirely, and we propose a solution by means of an algorithm due to Roune ([Roune, 2010]) which exploit the concepts of corners and Koszul simplicial complexes, originally introduced by Bayer ([Bayer, 1996]).

Let us see in more detail the main results of this thesis.

Let R := K[x1, . . . , xk]be a polynomial ring graded by a positive matrix

W ∈Matm,k(N) and let I be an ideal of R homogeneous respect to W . The

numerical function HR/I : Nm → N defined by HR/I(d) := dimK((R/I)d) is

called Hilbert function of R/I.

If the graduation on R is standard (i.e. W = [1, . . . , 1] ∈ Nk_{), H}
R/I is

definitively equal to a polynomial, called Hilbert polynomial of R/I, which has degree less or equal to k − 1.

We focus our attention on the less well-known case of non-standard grad-v

vi CONTENTS uation.

Let R be graded by W := [d1, . . . , dk] ∈ Nk+. The Hilbert function is

definitely equal to a quasi-polynomial PW

R/I := {P0, . . . , Pd−1} of period d,

where d := lcm(d1, . . . , dk), that is HR/I(n) = Pi(n) for all n sufficiently

large and n ≡ i (mod d).

We prove the existence of Hilbert quasi-polynomials and we give an ex-pository account of their construction. For this purpose, a particular power series associated to the Hilbert polynomial, called Hilbert-Poincaré series, plays a key role. It is defined by HPR/I(t) :=

P

n≥0HR/I(n)tn.

We show that PW

R/I is a quasi-polynomial with rational coefficients and,

for all i = 0, . . . , d − 1, if I = (0) then Pi has degree equal to k − 1, else Pi

has degree less or equal to k − 2. Anyway, for each Pi

lc (Pi) =

1 (k − 1)!Qk

i=1di

We restrict our study to (R/I, W ) with I = (0) and W = [d1, . . . , dk]

a weight vector such that gcd(d1, . . . , dk) = 1, in fact we can easily obtain

P_{R/I}W0 by P_{R}W, where W = _{gcd(d}01
1,...,d0k)

W0.

We present some proprieties of the coefficients of PW

R . Through an

anal-ysis of some numerical examples, we have noted some regularity for the coef-ficients of PW

R . In particular, we prove that PRW can be written as a sum of a

polynomial with rational coefficients of degree k − 1 and a quasi-polynomial with rational coefficients of degree δ − 1, where

δ := max{|I| | gcd(di)i∈I 6= 1and I ⊆ {1, . . . , k}}

This means that the coefficients of terms of degree δ, . . . , k − 1 are the same
for all Pi. Whereas, the r-th coefficient of P_{R}W has period equal to

δr := lcm( gcd(di)i∈I | |I| = r + 1and I ⊆ {1, . . . , k})

We devise a strategy to find the formulas for the coefficients of Hilbert quasi-polynomials and we compute two of them. If we denote the rth coef-ficient of PW

R by cr and if we suppose δ ≤ r , we have

ck−2 = Pk i=1di 2(k − 2)!Qk i=1di ck−3= 3(Pk i=1di)2−Pki=1d2i 24(k − 3)!Qk i=1di

In addition, we present an algorithm for the effective calculation of Hilbert quasi-polynomials. We have implemented it in SINGULAR, a free and

open-source computer algebra system for polynomial computations, and we show some numerical experiments.

CONTENTS vii In the last part of this thesis we consider an other related problem: for some graduations on R := K[x1, . . . , xk]given by W = (wij) ∈Matm,k(N),

the computation of Hilbert-Poincarè series associated to R/I is very expen-sive. For this reason, we present a way to get only a part of Hilbert-Poincaré series without computing it entirely. We use the following formula for the Hilbert-Poincaré series which exploits corners and Koszul simplicial com-plexes of I

H(I) := 1 + X

m∈cor(I)

χ(∆I_{m}) m

Here, χ denotes the Euler characteristic of a simplicial complex, ∆I m is the

Koszul simplicial complex of I at monomial m, and cor(I) is the set of corners of I. By substituting tw1i

1 · · · twmmi for xi in H(I), for all i = 1, . . . , k,

we obtain exactly the numerator f(t1, . . . , tm)of the Hilbert-Poincaré series.

If we want only a part of f(t1, . . . , tm), we stop the computation of corners

at one point, instead of computing all corners, in order to reduce the cost of computation.

This thesis expands Cirillo’s bachelor thesis ([Cirillo, 2014]), in which he shows only the existence of Hilbert quasi-polynomials and he gives a simple algorithm implemented in CoCoA to compute them.

This thesis is structured as follows:

Chapter 1: An overview of Hilbert function, Hilbert-Poincaré series and Hilbert polynomial of standard graded rings.

Chapter 2: Our results concerning Hilbert quasi-polynomial, such as exis-tence, degree, proprieties and formula for its coefficients.

Chapter 3: An algorithm for computing Hilbert quasi-polynomial.

Chapter 4: A strategy to get a part of Hilbert-Poincaré series without com-puting it entirely.

### Chapter 1

## Basic knowledge

In this chapter we recall basic definitions, as graded rings and graded modules, and some well known results concerning Hilbert function, Hilbert-Poincaré series and Hilbert polynomial associated to a graded module. Most of the material in this chapter is standard and we refer to [Robbiano, 2005] for the proofs and for further results.

In this work, all monoids and rings are assumed to be commutative.

### 1.1

### Graded rings and graded modules

In this section we introduce and study quite general kinds of gradings, namely graded rings and graded modules.

Definition 1.1Let (Γ, +) be a monoid. A ring R is called a graded ring
over Γ, or a Γ-graded ring, if there exists a family of additive subgroups
{R_{i}}_{i∈Γ} such that

• R = Li∈ΓRi,

• RiRj ⊆ Ri+j for all i, j ∈ Γ.

The elements of Ri are called homogeneous elements of degree i.

The subgroup Ri is called the homogeneous component of degree i of

R.

We note that, if R is a Γ-graded ring and 0 is the identity of R, then 0 is a homogeneous element of R of every degree. Moreover, the decomposition of every element into its homogeneous components is unique, since in the definition there is a direct sum.

2 CHAPTER 1. BASIC KNOWLEDGE The following two examples constitute the most important situations in which we shall meet graded rings.

Example 1.2. Let S be a ring, let n ≥ 1, and let R = S[x_{1}, . . . , xn] be a

polynomial ring over S. If we let

Ri := {f ∈ R | deg(t) = i for all t ∈ Supp(f)}

for every i ∈ N, we make R into an N-graded ring, where deg(xα1

1 · · · x αn

n ) := α1+ · · · + αn

This grading satisfies deg(x1) = · · · = deg(xn) = 1and is called the

stan-dard gradingof R. The elements of Ri are called homogeneous

polyno-mialsof degree i.

Example 1.3. Let S be a ring, let n ≥ 1, and let R = S[x1, . . . , xn] be

a polynomial ring over S. For each α := (α1, . . . , αn) ∈ Nn, we define

Rα := S · xα11· · · xnαn. In this way R = Lα∈NnRα become an Nn-graded

ring. We can also view R as a Zn_{-graded ring if we define R}

α = 0 for all

α = (α1, . . . , αn) ∈ Zn such that αi< 0 for some i ∈ {1, . . . n}.

Throughout the remainder of this chapter, R will be a Γ-graded ring, where (Γ, +) denotes a monoid.

Definition 1.4A R-module M is called a Γ-graded module over R, or a Γ-graded R-module, if there exists a family of subgroups {Md}d∈Γ such

that

• M = Ld∈ΓMd,

• RiMd⊆ Mi+d for all i, d ∈ Γ.

The elements of Mdare called homogeneous elements of degree d of M.

The submodule Md is called the homogeneous component of degree d

of M.

Definition 1.5An R-submodule N of the Γ-graded R-module M is called a Γ-graded R-submodule of M if we have N = Ld∈Γ(N ∩ Md).

A Γ-graded submodule of R is also called a Γ-homogeneous ideal of R. Let N ⊆ M be a Γ-graded R-submodule. We can equip the residue class module M/N with the structure of a Γ-graded R-module by defining

1.2. POLYNOMIAL RINGS GRADED BY MATRICES 3 (M/N )d:= Md/Ndfor every d ∈ Γ. In particular, the residue class ring R/I

of R by a homogeneous ideal I is again a Γ-graded ring.

Given a Γ-graded R-module M, there exists a notation to indicate that we have altered M by shifting its grading by γ steps.

Definition 1.6Let M be a Γ-graded module over a Γ-graded ring R with graduation given by M := {Md}d∈Γ and let γ ∈ Γ. Then, the family

{M0

d}d∈Γ := Md+γ is still a graduation of M and it is the graduation M

shifted by γ. The shift of the graduation M by γ is denoted by M(γ). Now we present the sets of morphisms between Γ-graded R-modules. Definition 1.7

(i) Let M, M0 _{be two Γ-graded R-modules. A morphism f : M → M}0 _{is}

called a morphism of degree γ ∈ Γ if f(Md) ⊆ Md+γ0 for all d ∈ Γ.

In particular, if γ = 0, f is called a homogeneous morphism. (ii) An exact sequence of graded modules is called graded of degree γ if

all its morphism are graded of degree γ. In particular, if γ = 0, it is called a homogeneous exact sequence.

### 1.2

### Polynomial rings graded by matrices

Having introduced the general notions of graded rings and modules, we are now going to concentrate on certain specific kinds of graded object. We are interested in gradings on polynomial rings which have two properties: the indeterminates are homogeneous and the constants are homogeneous of degree zero.

In what follows, let K be a field, n ≥ 1, and R := K[x1, . . . , xk]be a

polyno-mial ring over K.

Definition 1.8Let m ≥ 1 and let R be equipped with a Zm-grading such that K ⊆ R0 and the indeterminates x1, . . . , xk are homogeneous elements.

For j = 1, . . . , k, let (w1j, . . . , wmj) ∈ Zm be the degree of xj. The matrix

W = (wij) ∈ Matm,k(Z) is called the degree matrix of the grading. In

other words, the columns of the degree matrix are the degrees of the inde-terminates. The rows of the degree matrix are called the weight vectors of the indeterminates x1, . . . , xk.

4 CHAPTER 1. BASIC KNOWLEDGE Let R be graded by a matrix W ∈ Matm,k(Z), then the degree of a term

t = xα1

1 · · · x αk

k of R is given by degW(t) := W · (α1, . . . , αk)T. The vector

(α1, . . . , αk)T is usually denoted by log(t).

In addition, the set of homogeneous polynomials of degree d ∈ Zm_{is denoted}

by RW

d , or simply by Rd if it is clear which grading we are considering. A

polynomial f ∈ RW

d is also called homogeneous of degree d.

Example 1.9. Let R = K[x, y, z] be graded by the matrix W =1 2 4

0 1 0

and consider the polynomial f = x4_{y + yz. Then f is homogeneous of degree}

(6, 1), because W · log(x4y)T = W · log(yz)T = (6, 1).

In view of the following proposition, we shall henceforward assume tacitly that the degree matrix W has full rank.

Proposition 1.10Let R be graded by a matrix W ∈ Mat_{m,k}_{(Z), W 6= 0,}
and let W have Z-linearly dependent rows, and let W0 _{be a submatrix of W}

which consists of a maximal linearly independent set of rows. Denote the
number of rows of W0 _{by m}0_{, and let V ∈ Mat}

m,m0(Z) be the matrix such

that V · W0_{= W}_{. Then it holds R}W0

d0 = R W

V ·d for all d0∈ Zm

0

.

Our next goal is to find hypothesis which force the homogeneous com-ponents of R to be finite dimensional K-space vectors, in order to define the Hilbert function of R which describes exactly the dimension of homogeneous components of R as K-space vectors.

Definition 1.11Let m ≥ 1, let R be graded by a matrix W of rank m in Matm,k(Z), and let w1, . . . , wm be the rows of W . We say that the grading

on R given by W is of positive type if there exist a1, . . . , am ∈ Z such that

all entries of a1w1+ · · · + amwm are positive. In this case, we shall also say

that W is a matrix of positive type.

Example 1.12. The standard grading of R, defined in Example 1.2, is given
by (1, . . . , 1) ∈ Zk _{and it is of positive type.}

Proposition 1.13Let R be graded by a matrix W ∈ Matm,k(Z) of positive

type, and let M be a finitely generated graded R-module. Then (i) R0 = K

1.3. HILBERT FUNCTION AND HILBERT-POINCARÉ SERIES 5 (ii) dimK(Md) < ∞, for all d ∈ Zm.

Definition 1.14Let W ∈ Matm,k(Z) be a matrix of rank m. The grading

on R defined by W is called positive if no column of W is zero and the first non-zero element in each column is positive. In this case, we shall also say that W is a positive matrix.

Notice that if W is positive, then we have degW(xi) > 0 for i = 1, . . . , k,

and hence R+=Ld>0RdW = (x1, . . . , xk).

The following proposition says that positive gradings are of positive type and there exists a change of grading which transforms a grading of positive type into a positive one.

Proposition 1.15 Let R be graded by a matrix W ∈ Mat_{m,k}_{(Z)}
(i) If the grading defined by W is positive, it is of positive type.

(ii) If the grading defined by W is of positive type, then there exists a non-singular matrix V ∈ Matm(Z) such that the grading defined by

the matrix W0 _{= V · W} _{is positive and R}W0

V ·d = RWd for all d ∈ Zm.

From now on, we suppose R graded by a positive matrix W ∈ Matm,k(Z).

### 1.3

### Hilbert function and Hilbert-Poincaré series

We are going to present two strongly related notions: the Hilbert function and the Hilbert-Poincaré series of a finitely generated graded module, which measure the growth of the dimension of its homogeneous components. Definition 1.16Let M be a finitely generated graded R-module. The nu-merical function HW

M : Zm−→ N defined by

H_{M}W(d) := dim_{K}(Md)

is called the Hilbert function of M.

When the graduation given by W is clear from the contest, we denote simply Hilbert function by HM instead of HMW.

6 CHAPTER 1. BASIC KNOWLEDGE Proposition 1.17 (Basic properties of Hilbert functions)

Let M be a finitely graded R-module.

(i) For every d ∈ Zm_{, the Hilbert function of the shifted module M(d) is}

given by

HM (d)(i) = HM(d + i)

for all i ∈ Zm

(ii) Given finitely generated graded R-modules M0_{, M}00_{and a homogeneous}

short exact sequence of graded R-modules 0 → M0 _{→ M → M}00 _{→ 0}_{,}

it holds HM(i) = HM0(i) + H_{M}00(i)for all i ∈ Zm.

(iii) Given finitely many finitely generated R-modules M1, . . . , Mr, it holds

HM1⊕···⊕Mr(i) = HM1(i) + · · · + HMr(i)

for all i ∈ Zm_{.}

Remark 1.18. Let W = (wij) ∈Matm,k(Z) and (i1, . . . , im) ∈ Zm. Then the

value HR(i1, . . . , im)of the Hilbert function of R graded by W is the number

of solutions (α1, . . . , αk) ∈ Nk of the system of Diophantine equations

w11y1+ · · · + w1kyk = i1 w21y1+ · · · + w2kyk = i2 ... ... wm1y1+ · · · + wmkyk = im (1.1) in the indeterminates y1, . . . , yk.

In appendix A, we recall how to solve a system of linear Diophantine equations by means of Smith normal form. This method produces all in-teger solutions, from which we have to consider only the positive ones. In general, finding the solutions of the system (1.1) is not the best way to com-pute the value of HR(i1, . . . , im) for a given (i1, . . . , im) ∈ Zm. Admittedly,

the description of HR through the solutions of system (1.1) is not very

il-luminating with respect to the overall structure of this function. To get a deeper insight, we use the values of the Hilbert function as the coefficients of a suitable extended power series.

Actually, in the standard graded case, there is a simple formula for HR

(Proposition (1.21)). Whereas, for m = 1 the problem is equivalent to determine the formula for the number of partitions of an integer into elements of a finite set and we’ll examine this case closer in Chapter 3.

1.3. HILBERT FUNCTION AND HILBERT-POINCARÉ SERIES 7 Definition 1.19Let M be a finitely generated graded R-module. The ex-tended power series

HP_{M}W(t) := X

d∈Nm

H_{M}W(d)td∈ NJtK

where t = (t1, . . . , tm), is called the Hilbert-Poincaré series of M.

In combinatorial terms, the Hilbert-Poincaré series of M is the generating function of the sequence given by the Hilbert function.

When the graduation given by W is clear from the contest, we denote simply Hilbert-Poincaré series by HPM instead of HPMW.

Proposition 1.20 (Basic properties of Hilbert-Poincaré series) Let M be a finitely generated graded R-module.

(i) For every d = (d1, . . . , dm) ∈ Zm, the Hilbert-Poincaré series of the

shifted module M(d) is given by

HPM (d)(t1, . . . , tm) = t−d_{1} 1· · · t−dmmHPM(t1, . . . , tm)

(ii) Given finitely generated graded R-modules M0_{, M}00_{and a homogeneous}

short exact sequence of graded R-modules 0 → M0 _{→ M → M}00 _{→ 0}_{,}

it holds HPM(t) = HPM0(t) + HP_{M}00(t).

(iii) Given finitely many finitely generated R-modules M1, . . . , Mr, it holds

HPM1⊕···⊕Mr(t) = HPM1(t) + · · · + HPMr(t)

(iv) Given a homogeneous polynomial f ∈ R\{0}, it holds

HP_{M/f M}(t) = HPM(t) − td11· · · tmdmHPM/(0:M(f ))(t) (1.2)

where d = (d1, . . . , dm) ∈ Zm is the degree of f.

In particular, if f is a non-zerodivisor for M then M/(0 :M (f )) = M

and hence

HPM/f M(t) = (1 − td11· · · t dm

m )HPM(t) (1.3)

Equation (1.2) is know as main formula and it is a very useful tool to compute Hilbert-Poincaré series of M. We observe that, if I is a homoge-neous ideal of R and f ∈ R\I, the main formula allows us to compute the Hilbert-Poincaré series of R/(I, f) by means of the Hilbert-Poincaré series of R/I and R/(I : (f)), which are simpler to compute than HPR/(I,f ).

Let us compute Hilbert function and Hilbert-Poincaré series in the
sim-plest case M = R with W = (1, . . . , 1) ∈ Nk_{.}

8 CHAPTER 1. BASIC KNOWLEDGE Proposition 1.21 Let R = K[x1, . . . , xk]be a standard graded polynomial

ring. Then HR(i) = k+i−1_{n−1}

for every i ∈ N and HPR(t) = _{(1−t)}1 k

Proof (Hint). Given i ∈ N, the Hilbert function H_{R}(i) counts the number
of monomials in k variables of degree i, which is expressed by the binomial

k+i−1
n−1
. Then
HPR(t) =
X
i≥0
HR(i)ti =
X
i≥0
k + i − 1
n − 1
ti
and, since (1 − t)−k _{=}P
i≥0
k+i−1
n−1 ti, we are done.

Let us compute an actual Hilbert-Poincarè series for R graded by a generic matrix of positive type.

Example 1.22. Let R = K[x_{1}, x2]be graded by W =

1 2 0 1

. Then we have
R_{(i}_{1}_{,i}_{2}_{)}6= 0 if and only if i_{2} ≥ 0 and i_{1}− 2i_{2} ≥ 0. In these degrees we have
dim_{K}(R(i1,i2)) = 1. Therefore we obtain

HPR(t1, t2) =
X
i2≥0
X
i1−2i2≥0
ti1
1t
i2
2 =
P
i2≥0t
2i1
1 t
i1
2
1 − t1
= 1
(1 − t1)(1 − t2_{1}t2)

This example suggests that there might be a simple formula for the Hilbert-Poincaré series of a polynomial ring. Next theorem delivers a very satisfactory answer.

Theorem 1.23 Let R = K[x1, . . . , xk]be graded by a positive matrix W =

(wij) ∈Matm,k(Z). Then HPR(t1, . . . , tm) = 1 Qk j=1(1 − t w1j 1 · · · t wmj m ) (1.4) Proof. We use induction on k, the number of variables of R.

For k = 1, we have degW(xi1) = (iw11, . . . , iwm1). Therefore we obtain

HPR(t1, . . . , tm) =
X
i≥0
tiw11
1 · · · t
iwm1
m =
1
1 − tw_{1}1j· · · twmj
m

and then the formula holds.

Now consider the case k > 1. Since degW(xk) = (w1k, . . . , wmk), then

Equa-tion (1.3) yields

HPR/(xk)(t) = (1 − t

w1k

1.4. HILBERT-SERRE THEOREM AND HILBERT POLYNOMIAL 9 Now we use the fact R/(xk) ∼= K[x1, . . . , xk−1]and the induction hypothesis

to finish the proof.

Equation (1.4) can be written concisely in the following way:
HPR(t) =
1
Qk
j=1(1 − tdegW(xj))
where tdegW(xj) _{= t}w1j
1 · · · t
wmj
m .

### 1.4

### Hilbert-Serre theorem and Hilbert polynomial

Let us focus definitively our attention on the case M = R/I, where I is a homogeneous ideal of R. Actually, in view of the next proposition we might consider only monomial ideals.

Before all, recall some preliminary notions.

We say that a total ordering σ on the terms of a ring R is a term-order, and we write σ ∈ TO(R), if t >σ 1 for all terms t of R.

We denote the ideal generated by the leading terms of polynomials in I by LTσ(I), namely LTσ(I) := (ltσ(f )|f ∈ I).

Proposition 1.24(Macaulay’s Lemma) Let I be a homogeneous ideal of a ring R and σ ∈ TO(R). Then

HR/I(d) = HR/LTσ(I)(d) for all d ∈ Z

m _{and} _{HP}

R/I = HPR/LTσ(I).

Proof. If we prove that dim_{K}((R/I)d) = dimK((R/LTσ(i))d)for all d ∈ Zm,

then the first part of the statement follows.

For a graded piece (R/I)d, we have a vector space basis given by {f1, . . . , fj}.

We can assume that lt(f1) >σ · · · >σ lt(fj). If they don’t span (R/LTσ(I))d,

pick m ∈ (R/LTσ(I))d which is not in the span, such that lt(m) = m0 is

minimal with respect to the term order σ. Of necessity, there is a polynomial g ∈ R/I with lt(g) = m0. Since g is a K-linear combination of the fi, m0

must be one the lt(fi) (since no cancellation can occur), contradicting the

choice of m0 _{minimal.}

The second part of the statement follows in a banal way by the first. We notice that in the proof of Macaulay’s lemma choice of term-order σ is irrelevant.

10 CHAPTER 1. BASIC KNOWLEDGE Next theorem gives us a characterisation of the Hilbert-Poincarè series HPR/I, in fact it shows that HPR/I is a rational function.

Theorem 1.25 (Hilbert-Serre) Let R := K[x_{1}, . . . , xk] be a polynomial

ring graded by a matrix W = (wij) ∈Matm,k(Z) of positive type and let I

a homogeneous ideal of R. Then there exists a polynomial f ∈ Z[t1, . . . , tm]

such that HPR/I(t1, . . . , tm) = f (t1, . . . , tm) Qk j=1(1 − t w1j 1 · · · t wmj m )

Proof. By Macaulay’s lemma, we can suppose I is a monomial ideal and let {p1, . . . , ph} be a minimal monomial system of generators for I, i.e. I =

(p1, . . . , ph). We denote the ideal (p1, . . . , ph−1)by J and by the main formula

we get

HPR/I(t) = HPR/(J,ph)(t) = HPR/J− t

degW(ph)_{HP}

R/(I:(ph))(t)

We observe that J and (I : (ph))have a minimal system of generators smaller

than that of I, so we can reduce the computation of the Hilbert-Poincaré
series of R/I to that of two ideals with a less number of generators. We can
interact this argument so that the resulting recursive algorithm to compute
HP_{R/I} finishes after a finitely number of steps and it yields a polynomial in
tmultiplied by HPRwhich is equal to Qk 1

j=1(1−t w1j 1 ···t

wmj

m ), then the statement

follows.

The polynomial f(t1, . . . , tm)which appears in the denominator of HPR/I

is called h-vector and it is denote by < I >W _{or simply < I >.}

In the proof of Hilbert-Serre theorem we have briefly illustrated a pro-cedure to compute the h-vector. Before applying it to a concrete case, we make a list of some useful rules which derive from the main formula.

Let xα_{= x}α1
1 · · · x
αk
k be a term of R. Then
(a) < 0 > = 1;
(b) < 1 > = 0;
(c) < xα_{> = 1 − t}deg_{W}(xα_{)}
;
(d) < xα_{I > = 1 − t}degW(xα)_{+ t}degW(xα) _{< I >};
(e) < I, xα_{> = < I > −t}degW(xα)_{< I : (x}α_{) >};

1.4. HILBERT-SERRE THEOREM AND HILBERT POLYNOMIAL 11 Example 1.26. Let R = [x, y, z] be graded by W = 1 1 2

0 3 1

, and let
I = (x2y, y3, z2). We want to compute the Hilbert-Poincaré series of R/I
following the algorithm sketched in the proof of Hilbert-Serre theorem.
Let us consider I = (J, z2_{), where J = (x}2_{y, y}3_{). Then, by rule (e) we get}

< I > = < J > −tdegW(z2)_{< J : (z}2_{) >}

Then we have to compute < J > and < J : (z2_{) >}_{. Start from the }

compu-tation of < J >. Let us consider J = (J0_{, y}3_{)}_{, where J}0 _{= (x}2_{y)} _{and again}

by the rule (e) we get

< J > = < x2y > −tdegW(y3)_{< J}0 _{: (y}3_{) >}

By rule (c), we have < x2_{y >= 1 − t}degW(x2y), and, since J0 _{: (y}3_{) = (x}2_{)},

< J0 : (y3) >= 1 − tdegW(x2). Then

< J > = (1 − tdegW(x2y)_{) − t}degW(y3)_{(1 − t}degW(x2)_{)}

= (1 − t3_{1}t3_{2}) − t3_{1}t9_{2}(1 − t2_{1})

Let us compute < J : (z2_{) >}_{. We observe that < J : (z}2_{) > = < J >}_{, then}

we have found
< I > = (1 − tdegW(z2)_{) < J > = (1 − t}4
1t22)[(1 − t31t32) − t31t92(1 − t21)]
Finally, we have
HPR/I =
< I >

(1 − tdegW(x))(1 − tdegW(y))(1 − tdegW(z))

= −t

9

1t112 + t71t112 + t71t52+ t51t92− t41t22− t13t92− t31t32

(1 − t1)(1 − t1t3_{2})(1 − t2_{1}t2)

Remark 1.27. By recalling that for all t = (t1, . . . , tm) and for all α ∈ Nm it

holds

(1 − tα)−1 =X

i≥0

(tα)i

it follows that the computation of HR/I from HPR/I is quite easy. In fact, it

suffices to invert the denominator and multiply the result by the numerator to get the canonical representation of the Hilbert-Poincaré series from which it is immediate to read the Hilbert function.

The main corollary of Hilbert-Serre theorem is the next result which shows that, under certain hypothesis, there exists a unique polynomial PR/I(n)

12 CHAPTER 1. BASIC KNOWLEDGE Corollary 1.28Let R be a polynomial standard graded ring and let I a homogeneous ideal of R. There exists a polynomial PR/I(t) ∈ Q[t] such that

H_{R/I}(n) = P_{R/I}(n)for all sufficiently large integer n 0.
Proof. By Hilbert-Serre theorem, we have

HPR/I(t) =

f (t) (1 − t)r =

h(t) (1 − t)s

where h(t) ∈ Z[t] is a polynomial such that h(1) 6= 0 and s ≤ r. Let
h(t) =
N
X
n=0
antn and (1 − t)−s=
∞
X
n=0
n + s − 1
s − 1
tn,
then
HP_{R/I}(t) =X
n≥0
" _{N}
X
i=0
ai
n + s − i − 1
s − 1
#
tn

So, for n ≥ N, we have

HR/I(n) = N X i=0 ai n + s − i − 1 s − 1

which is a polynomial in n with leading term h(1) (s−1)!n

s−1 _{6= 0}_{.}

The polynomial PR/I(t)defined in the corollary 1.28 is called the Hilbert

polynomial of R/I.

Aim of next chapter is to show the existence of a sort of Hilbert poly-nomial for rings graded by a weight vector with entries not necessarily all equal to 1. Before treating this case, we make some remarks about Hilbert polynomial.

By the proof of corollary 1.28, we get that PR/I(t)is a polynomial with

rational coefficients and, if the Hilbert-Poincaré series is
HPR/I(t) =
h(t)
(1 − t)s =
a0+ a1t + . . . aNtN
(1 − t)s
with h(1) 6= 0, we have
P_{R/I}(t) =
N
X
i=0
ai
t + s − i − 1
s − 1

1.4. HILBERT-SERRE THEOREM AND HILBERT POLYNOMIAL 13 formula for PR/I(t). In addition, the degree of PR/I(t)is s−1 and its leading

coefficient is h(1) (s−1)!.

Example 1.29. Let R = K[x, y, z] and let I = (x2y, xy2, z3)be an ideal of R. The Hilbert-Poincaré series of R/I is

HP_{R/I}(t) = 1 + 2t + 3t

2_{+ t}3_{− t}5

1 − t

Then, by using the formula above with s = 1, N = 5, a0 = 1, a1 = 2, a2 = 3,

a3= 1, a4 = 0 and a5 = −1, we get PR/I(t) = 5 X i=0 ai t + 1 − i − 1 1 − 1 = 5 X i=0 ai= 6

for all t > 5. Thus, we have

HR/I(n) = 1 if n = 1 3 if n = 2 5 if n = 3 6 if n = 4 5 if n = 5 6 if n ≥ 6

### Chapter 2

## Hilbert quasi-polynomial

In this chapter, we consider a polynomial ring R over an infinite field K of
characteristic 0 where the degrees of the variables are assumed to be positive
integers with no further restriction. We prove that the Hilbert function of
R/Iis definitely equal to a quasi-polynomial P_{R/I}W , where I is a homogeneous
ideal of R, and we investigate the structure of PW

R/I, such as its degree, its

leading coefficient and proprieties of its coefficients.

### 2.1

### Existence of the Hilbert quasi-polynomial

From now on, (R, W ) stands for a polynomial ring R := K[x1, . . . , xk]

with the graduation given by W := [d1, . . . , dk] ∈ Nk+. Let I be a

homoge-neous ideal of (R, W ). Thanks to Macaulay’s lemma, we can suppose that I is a monomial ideal. In addition, we denote the least common multiple of d1, . . . , dk by d. We are going to prove that the Hilbert function of (R/I, W )

is definitely equal to a quasi-polynomial of period d.

Definition 2.1 A function f : N → N is a quasi-polynomial of period s if there exist polynomials p0, . . . , ps−1 ∈ Q[x] such that f(n) = pi(n) when

n ≡ i mod s.

Proposition 2.2 Let (R, W ) and I be as above, there exists a uniquely determined quasi-polynomial PW

R/I of period d such that HR/I(n) = P W R/I(n)

for all n 0.

Proof. From the Hilbert-Serre theorem, we know that the Hilbert-Poincaré series of R/I can be written as a rational function

HPR/I(t) = h(t) Qk i=1(1 − tdi) ∈ Q[|t|] 15

16 CHAPTER 2. HILBERT QUASI-POLYNOMIAL Let ζ be a primitive dth root of unity, so we can write

k
Y
i=1
(1 − tdi_{) =}
d−1
Y
j=0
(1 − ζjt)αj_{,} for some α
j ∈ N such that
d−1
X
j=0
αj =
k
X
i=1
di

By using this relation and partial fractions, we have
HP_{R/I}(t) = h(t)
Qd−1
j=0(1 − ζjt)αj
=
d−1
X
j=0
Qj(t)
(1 − ζj_{t)}αj (2.1)

where Qj(t) ∈ C[t] is a polynomial of degree nj for all j = 0, . . . , d − 1,

namely Qj(t) =Pn_{h=0}j ajhth.

We are going to consider the addends individually, so
Qj(t)
(1 − ζj_{t)}αj =
nj
X
h=0
ajhth
!
· 1
(1 − ζj_{t)}αj
=
nj
X
h=0
ajhth
!
·
X
n≥0
n + αj − 1
n
ζjntn
=X
n≥0
"nj
X
h=0
ajh
n + αj− h − 1
n − h
ζj(n−h)
#
tn
=X
n≥0
"nj
X
h=0
ajh
n + αj− h − 1
αj− 1
ζj(n−h)
#
tn=X
n≥0
bjntn
where
bjn:=
nj
X
h=0
ajh
n + αj− h − 1
αj− 1
ζj(n−h)

We can rewrite the Hilbert-Poincaré series in the following way:
HP_{R/I}(t) =
d−1
X
j=0
Qj(t)
(1 − ζj_{t)}αj =
d−1
X
j=0
X
n≥0
bjntn=
X
n≥0
d−1
X
j=0
bjn
tn

and, therefore, for all n ≥ max

j {nj}, we have HR/I(n) = d−1 X j=0 bjn= d−1 X j=0 nj X h=0 ajh n + αj− h − 1 n − h ζj(n−h)= d−1 X j=0 Sj(n)ζjn where Sj(n) := nj X h=0 ajh n + αj− h − 1 n − h ζ−jh.

2.1. EXISTENCE OF THE HILBERT QUASI-POLYNOMIAL 17 So, given n ≥ max

j {nj}, let i be an integer such that 0 ≤ i ≤ d − 1 and

i ≡ n mod d, the Hilbert function evaluated at n is equal to the evaluation at n of the polynomial

Pi(x) := S0(x) + S1(x)ζi+ · · · + Sd−1(x)ζ(d−1)i.

The quasi-polynomial PW

R/I = {P0, . . . , Pd−1} defined in proposition 2.2 is

called the Hilbert quasi-polynomial of (R/I,W).

By the proof, we find out that HR/I(n) = P_{R/I}W (n)for all n ≥ max
j {nj},

where nj are the degrees of the polynomials Qj(t)which appear in Equation

(2.1) of the Hilbert-Poincaré series.

Remark 2.3. Proposition 2.2 asserts that the Hilbert quasi-polynomial has period equal to d, but it doesn’t assure that the d polynomials are necessarily all distinct. We’ll see in next examples that it can happen.

In addition, in the proof of proposition we yield a way to compute the Hilbert quasi-polynomial of (R/I, W ), when the Hilbert-Poincaré series is known.

Recall that, thanks to the corollary 1.28, if R is a polynomial ring with standard graduation and I is a homogeneous ideal of R, then the Hilbert polynomial PR/I is a polynomial with rational coefficients and PR/I(n) ∈ N

for all sufficiently large n ∈ N. We’ll show that the Hilbert quasi-polynomial of (R/I, W ) satisfies the same proprieties. Before all, we prove a general result concerning polynomials and their coefficients.

Theorem 2.4 Let f(x) ∈ C[x] be a polynomial of degree n. If f(xi) ∈ Z

for at least n + 1 points xi ∈ Z, then f(x) ∈ Q[x].

Proof. Consider f(x) = a_{n}xn+ · · · + a1x + a0. By hypothesis, there exist

x0, . . . , xn ∈ Z such that f(xi) = yi ∈ Z. If we substitute the equation of

f (x)in here, we get a system of linear equations in the coefficients ak. The

system in matrix-vector form reads
xn_{0} xn−1_{0} xn−2_{0} . . . x0 1
xn_{1} xn−1_{1} xn−2_{1} . . . x1 1
... ... ... ... ...
xn_{n} xn−1_{n} xn−2_{n} . . . xn 1
an
an−1
...
a0
=
y0
y1
...
yn

Let V be the matrix on the left and let b be the vector [y0, y2, . . . , yn]T.

18 CHAPTER 2. HILBERT QUASI-POLYNOMIAL f (x)are uniquely determined by

[an, an−1, . . . , a0]T = V−1b

Since b ∈ Zn+1 _{and V ∈ Mat}

n+1(Z), it holds V−1 ∈ Matn+1(Q) and the

statement follows.

Proposition 2.5Let (R, W ) and I be as above and let P_{R/I}W = {P0, . . . , Pd−1}

be the Hilbert quasi-polynomial of (R/I, W ). Then Pi(x) ∈ Q[x] for all

i = 0, . . . , d − 1.

Proof. By definition, Pi(n) ∈ N for all sufficiently large n ∈ N such that

i ≡ n mod d, then the hypothesis of the theorem 2.4 are satisfied and we can conclude that PW

R/I is a quasi-polynomial with rational coefficient.

Thanks to the following propositions, we can restrict in a particular case: (R/I, W ) with I = (0) and W = [d1, . . . , dk] a weight vector such that

gcd(d1, . . . , dk) = 1. In fact, for any homogeneous ideal I of R we can

com-pute the Hilbert quasi-polynomial PW

R/Istarting from PRW, and given a weight

vector W0 _{:= [d}0

1, . . . , d0k]such that c := gcd(d01, . . . , d0k) 6= 1, we can compute

the Hilbert quasi-polynomial PW0

R starting from PRW, with W := 1 cW

0_{.}

Proposition 2.6Let P_{R}W be the Hilbert quasi-polynomial of (R, W ). Let I
be a homogeneous ideal of R and let HPR/I(t) = Qk h(t)

i=1(1−tdi)

be the Hilbert-Poincaré series of R/I, where h(t) := Pr

j=0ajtj. Then, for all n sufficiently

large, the Hilbert quasi-polynomial PW

R/I is given by
P_{R/I}W (n) =
r
X
j=0
ajPRW(n − j)
Proof. Since
HPR(t) =
X
n≥0
HR(n)tn=
1
Qk
i=1(1 − tdi)
we have
HP_{R/I}(t) = h(t)
Qk
i=1(1 − tdi)
=
X
n≥0
HR(n)tn
r
X
j=0
ajtj
=X
n≥0
r
X
j=0
ajHR(n − j)
tn

2.1. EXISTENCE OF THE HILBERT QUASI-POLYNOMIAL 19 with the convention HR(n) = 0 for all n < 0. Therefore, for all n ≥ r,

HR/I(n) =

Pr

j=0ajHR(n − j) . Given that HR(n) = PRW(n) for all

suffi-ciently large n, then HR(n − j) = P_{R}W(n − j) for all sufficiently large n and

so HR/I(n) =

Pr

j=0ajPRW(n − j) for all sufficiently large n. The statement

follows.

Proposition 2.7Let P_{R}W = {P0, . . . , Pd−1}be the Hilbert quasi-polynomial

of (R, W ). If W0 _{:= a · W = [d}0

1, . . . , d0k] for some a ∈ N+, then the Hilbert

quasi-polynomial PW0

R = {P00, . . . , Pad−10 } of (R, W0) is such that:

(i) If a doesn’t divide i, then P0 i = 0 ;

(ii) If a divides i, then P0

i(x) = Pi a(

x a) .

Proof.

(i) Let i be a positive integer such that a doesn’t divide i. We observe that
H_{R}W0(n) = 0for all n ∈ N such that n ≡ i mod a. Indeed, assume that
H_{R}W0(n) 6= 0 for some n ≡ i mod a. Then there exist a1, . . . , ak ∈ N

such that a1d01+ · · · + akd0k = n. By hypothesis, a divides d 0

1, . . . , d0k,

hence a divides n and so a divides i, which is a contradiction. Since
P_{i}0(n) = H_{R}W0(n)for all sufficiently large n such that n ≡ i mod a, then
P_{i}0 has an infinitive number of roots and so P_{i}0= 0.

(ii) Recall that for all sufficiently large n such that n ≡ i mod a, we have
Pi(n) = HRW(n) = #{(a1, . . . , ak) ∈ Nk|a1d1+ . . . akdk= n}
Then,
P_{i}0(n) = H_{R}W0(n) = #{(b1, . . . , bk) ∈ Nk | ab1d1+ · · · + abkdk= n}
= #{(b1, . . . , bk) ∈ Nk | b1d1+ · · · + bkdk=
n
a}
= H_{R}W(n
a) = Pai(
n
a)

To avoid endlessly repeating these hypothesis, we shall use W to denote a weight vector [d1, . . . , dk] such that gcd(d1, . . . , dk) = 1, throughout the

20 CHAPTER 2. HILBERT QUASI-POLYNOMIAL

### 2.2

### Degree and leading coefficient of the Hilbert

### quasi-polynomial

In this section, we investigate general proprieties of the Hilbert quasi-polynomial of (R, W ).

First of all, we introduce the following notation which will help us to rewrite the denominator of the Hilbert-Poincaré series of (R, W ) in a suit-able way.

Given d1, . . . , dk∈ N+, with d := lcm(d1, . . . , dk), we define

• δ := max{|I| | gcd(di)i∈I 6= 1and I ⊆ {1, . . . , k}},

• ˆds:= _{d}d_{s},
• Ms := { ˆds, 2 ˆds, . . . , (ds− 1) ˆds} for all s = 1, . . . , k,
• Tr:=
[
J ⊆{1,...,k}
|J|=r
(\
s∈J
Ms) for all r = 1, . . . , k.

Observe that Tr is the set of the elements p ∈ Sk_{i=1}Mi which belong to

exactly r sets Mi.

Now we’ll analyse the denominator of the Hilbert-Poincaré series of (R, W ), which we denote by g(t). Let ζ := ζd be a primitive dth root of unity. We

remark that ζdj = ζ

ˆ

dj for all j = 1, . . . , k. Then we can write

g(t) =
k
Y
i=1
(1 − tdi_{) = (1 − t)}k
k
Y
i=1
h
(1 − ζdˆi_{t) · · · (1 − ζ}(di−1) ˆdi_{t)}
i
= (1 − t)k Y
j∈M1
(1 − ζjt) · · · Y
j∈Mk
(1 − ζjt)
= (1 − t)k Y
j∈Tk
(1 − ζjt)k· · · Y
j∈T1
(1 − ζjt)
(2.2)

Example 2.8. Let W = [2, 3, 4]. By using the notation introduced above, we have d = 12, δ = 2, ˆd1 = 6, ˆd2= 4 and ˆd3 = 3.

2.2. DEGREE AND LEADING COEFFICIENT 21 T2 = {6}and T3= ∅.

Let ζ := ζ12, then we can rewrite g(t) in the following way:

g(t) = (1 − t2)(1 − t3)(1 − t4)

= (1 − t)3(1 − ζ6_{t) (1 − ζ}4_{t)(1 − ζ}8_{t) (1 − ζ}3_{t)(1 − ζ}6_{t)(1 − ζ}9_{t)}
= (1 − t)3(1 − ζ6t)2 (1 − ζ3t)(1 − ζ4t)(1 − ζ8t)(1 − ζ9t)

We are now going to show some lemmas concerning the sets Ms and Tr.

Lemma 2.9Let d_{1}, . . . , dk ∈ N+. For any subset {j1, . . . , jr} ⊆ {1, . . . , k},

it holds Mj1 ∩ · · · ∩ Mjr = d gcd(dj1, . . . , djr) , 2 · d gcd(dj1, . . . , djr) , . . . . . . , (gcd(dj1, . . . , djr) − 1) · d gcd(dj1, . . . , djr)

Proof. We use induction on r, the number of intersected subsets.

The case r = 1 is trivial. Suppose r ≥ 2 and consider, without loss of generality, the subsets M1, . . . , Mr instead of Mj1, . . . , Mjr. By induction

hypothesis, we have M1∩ · · · ∩ Mr−1 = d gcd(d1, . . . , dr−1) , 2 · d gcd(d1, . . . , dr−1) , . . . . . . , (gcd(d1, . . . , dr−1) − 1) · d gcd(d1, . . . , dr−1)

We observe that M1∩ · · · ∩ Mr−1 has the same structure of Ma, with a :=

gcd(d1, . . . , dr−1)(we can suppose that a is one of the given integers, without

undermining the statement). Therefore M1∩ · · · ∩ Mr = Ma∩ Mr and, by

using again the induction hypothesis, we obtain M1∩ · · · ∩ Mr = d gcd(a, dr) , 2 · d gcd(a, dr) , . . . , (gcd(a, dr) − 1) · d gcd(a, dr)

and since a = gcd(d1, . . . , dr−1), we are done.

22 CHAPTER 2. HILBERT QUASI-POLYNOMIAL Proof. We assume that Th 6= ∅ for some h = δ + 1, . . . , k. Consider p ∈ Th

and suppose, without loss of generality, that p ∈ M1∩ · · · ∩ Mh. By lemma

2.9, p = p1·_{gcd(d}_{1}d_{,...,d}_{h}_{)} with 1 ≤ p1 ≤ gcd(d1, . . . , dh) − 1. Since δ < h, we

must have gcd(d1, . . . , dh) = 1. Therefore p is a multiple of d, which is a

contradiction because of p < d.

By lemma 2.10, in Equation (2.2) the set Tδ+1, . . . , Tk are empty, so

ultimately we have
g(t) =
k
Y
i=1
(1 − tdi_{) = (1 − t)}k
Y
j∈Tδ
(1 − ζjt)δ
· · ·
Y
j∈T1
(1 − ζjt)
(2.3)
and we obtain the following useful expression for the Hilbert-Poincaré
series
HPR(t) =
1
g(t) =
Q0(t)
(1 − t)k +
X
j∈Tδ
Q(δ)_{j} (t)
(1 − ζj_{t)}δ + · · · +
X
j∈T1
Q(1)_{j} (t)
(1 − ζj_{t)} (2.4)

Remark 2.11. We have already seen in the proof of proposition 2.2 that an addend Qj(t)

(1−ζj_{t)}αj of HPR(t), with Qj(t) =Pn_{h=0}j ajhth, can be written in the

following way
Qj(t)
(1 − ζj_{t)}αj =
X
n≥0
"nj
X
h=0
ajh
n + αj− h − 1
αj− 1
ζj(n−h)
#
tn (2.5)
Then the rational function Qj(t)

(1−ζj_{t)}αj corresponds to a power series in t where

the nth coefficient depends on n in a polynomial way through the binomial

n+αj−h−1

αj−1

and its degree in n is αj− 1.

We consider the equation (2.4) of the Hilbert-Poincaré series and, by using partial fractions, we get

HPR(t) =
1
g(t) =
A1
1 − t +
A2
(1 − t)2 + · · · +
Ak
(1 − t)k+
+ X
j∈Tδ
B_{j,1}(δ)
1 − ζj_{t} + · · · +
B(δ)_{j,δ}
(1 − ζj_{t)}δ
+ · · · +
X
j∈T1
"
B_{j,1}(1)
1 − ζj_{t}
#
(2.6)

2.2. DEGREE AND LEADING COEFFICIENT 23
for some Ai, B(r)_{j,h} ∈ Q.

Remark 2.12. By multiplying both sides of Equation (2.6) by g(t), we get

1 =
k
X
t=1
At(1 − t)k−t·
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
+
+ X
j∈Tδ
(1 − t)k·
δ−1
Y
i=1
Y
r∈Ti
(1 − ζjt)i
·
k
X
m=1
B_{j,m}(δ) · Y
s∈Tδ
s6=j
(1 − ζst)δ−m
+
+
...
+
+ X
j∈T1
B_{j,1}(1)(1 − t)k· Y
r∈Tδ
(1 − ζrt)δ· · · Y
r∈T2
(1 − ζr)2 Y
s∈T1
s6=j
(1 − ζst)
(2.7)

Remark 2.13. By Equation (2.6), we have

Q0(t) = A1(1 − t)k−1+ A2(1 − t)k−2+ · · · + Ak (2.8)

Lemma 2.14 The constant A_{k} which appears in Equation (2.6) and (2.8)
is such that
Ak=
1
Qk
i=1di
.

Proof. By evaluating Equation (2.7) at t = 1, we get
1 = Ak·
Y
j∈Tδ
(1 − ζj)δ· · · Y
j∈T1
(1 − ζj)
If we prove that Qj∈δ(1 − ζj)δ· · ·
Q
j∈T1(1 − ζ
j_{)} _{is equal to Q}k
i=1di, we are

24 CHAPTER 2. HILBERT QUASI-POLYNOMIAL which derives from equation (2.3):

g(t)
(1 − t)k =
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
g(t)
(1 − t)k =
k
Y
j=1
1 − tdj
1 − t =
k
Y
j=1
(1 + t + · · · + tdj−1_{)}
Then, it holds
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt) =
k
Y
j=1
(1 + t + · · · + tdj−1_{)} (2.9)

By evaluating both sides of Equation (2.9) at t = 1, the statement follows.

Proposition 2.15Let (R, W ) and P_{R}W be as above. The degree of Pi is

equal to k − 1 for all i = 0, . . . , d − 1 and its leading coefficient lc(Pi)is such

that

lc(Pi) =

1 (k − 1)!Qk

i=1di

Proof. Since δ < k by hypothesis and thanks to Equation (2.4) and remark 2.11, it suffices to analyse the term

Q0(t)
(1 − t)k =
X
n≥0
"_{k−1}
X
h=0
ah
n − h + k − 1
k − 1
#
tn

where ah:= a0,h for all h = 0, . . . , k − 1.

We observe that Pk−1

h=0ah n−h+k−1_{k−1} = S0(n)and then the degree of Pi

is less or equal to k − 1. It remains to show that the coefficient of nk−1 _{is}

not 0. In particular, we wish to show that it is equal to 1 (k−1)!Qk

i=1di.

Suppose n ≥ k − 1, then the nth coefficient of HPR(t)is equal to

S0(n) = k−1 X h=0 ah n − h + k − 1 k − 1 = k−1 X h=0 ah (n − h + k − 1)(n − h + k − 2) · · · (n − h + 1) (k − 1)! = k−1 X h=0 ah· nk−1

2.2. DEGREE AND LEADING COEFFICIENT 25 Then the leading coefficient of S0(n)is

1 (k − 1)! k−1 X h=0 ah = Q0(1) (k − 1)! = Ak (k − 1)! = 1 (k − 1)!Qk i=1di as required.

Now, we’ll give a degree bound for the Hilbert quasi-polynomial of (R/I, W ), when I 6= (0).

Proposition 2.16Let I 6= (0) a homogeneous ideal of R and let P_{R/I}W the
Hilbert quasi-polynomial of (R/I, W ). Then the degree of Pi is less or equal

to k − 2.

In order to prove Proposition 2.16, we need the following results.

Let I 6= (0) be a homogeneous monomial ideal and consider the Hilbert-Poincaré series HPR/I(t) = Qk f (t)

i=1(1−tdi)

= f (t)_{g(t)} of (R/I, W ).

Let f(t) := q(t)g(t) + r(t), where q(t), r(t) ∈ Q(t) and deg r < deg g. Then, we can write

HPR/I(t) =
f (t)
g(t) = q(t) +
r(t)
g(t) = q(t) +
C1
1 − t+
C2
(1 − t)2 + · · · +
Ck
(1 − t)k+
+X
j∈Tδ
D_{j,1}(δ)
1 − ζj_{t}+ · · · +
D(δ)_{j,δ}
(1 − ζj_{t)}δ
+ · · · +
X
j∈T1
"
D_{j,1}(1)
1 − ζj_{t}
#
(2.10)
for some Ci, D_{j,h}(r)∈ Q.

Lemma 2.17 The constant Ck which appears in Equation (2.10) is zero.

26 CHAPTER 2. HILBERT QUASI-POLYNOMIAL
r(t) =
k
X
t=1
Ct(1 − t)k−t·
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
+
+X
j∈Tδ
(1 − t)k·
δ−1
Y
i=1
Y
r∈Ti
(1 − ζjt)i
·
k
X
m=1
D_{j,m}(δ) · Y
s∈Tδ
s6=j
(1 − ζst)δ−m
+
+
...
+
+X
j∈T1
D_{j,1}(1)(1 − t)k· Y
r∈Tδ
(1 − ζrt)δ· · · Y
r∈T2
(1 − ζr)2 Y
s∈T1
s6=j
(1 − ζst)
(2.11)
By evaluating Equation (2.11) at t = 1, we have

r(1) = Ak· Y j∈T δ (1 − ζj)δ· · · Y j∈T1 (1 − ζj)

We already know that Q_{j∈T δ}(1 − ζj)δ· · ·Q

j∈T1(1 − ζ

j_{) =}Qk

i=1di, then we

have to show that r(1) = 0. Actually, it suffices to prove that f(1) = 0, because f(1) = q(1)g(1) + r(1) and f(1) = g(1) = 0 imply r(1) = 0.

We prove that f(1) = 0 by induction on the cardinality s of a minimal set of generators of I.

Suppose s = 1, since I 6= (0) there exists m ∈ I such that I = (m). Then f (t) =< I >= 1 − tdegW(m), so obviously f(1) = 0.

If s ≥ 2, let {m1, . . . , ms} a minimal set of generators of I, then I =

(m1, . . . , ms) and, by using the main formula, we have

f (t) = < (m1, . . . , ms) > =

< (m1, . . . , ms−1) > −tdegW(ms)< (m1, . . . , ms−1) : (ms) >

and we can conclude by induction hypothesis.

We are now able to demonstrate that for all i = 0, . . . , d − 1 it holds deg(Pi) ≤ k − 2.

2.2. DEGREE AND LEADING COEFFICIENT 27 Proof of Proposition 2.16. We consider the following expression for the Hilbert-Poincaré series, which we have already seen in Equation (2.10)

HP_{R/I}(t) = f (t)
g(t) = q(t) +
r(t)
g(t) = q(t) +
C1
1 − t+
C2
(1 − t)2 + · · · +
Ck
(1 − t)k+
+X
j∈Tδ
D_{j,1}(δ)
1 − ζj_{t}+ · · · +
D(δ)_{j,δ}
(1 − ζj_{t)}δ
+ · · · +
X
j∈T1
"
D_{j,1}(1)
1 − ζj_{t}
#
Since δ < k, and by lemma 2.17, which asserts that Ck= 0, we have that the

maximum exponent in the denominator of HPR/I is at most k−1. Therefore,

by recalling the construction of Hilbert quasi-polynomial done in the proof
of proposition 2.2, we write
HPR/I(t) =
d−1
X
j=0
Qj(t)
(1 − ζj_{t)}αj

for some polynomials Qj(t) := P nj

h=0ajhth ∈ C[t] of degree nj and with

αj ≤ k − 1 for all j = 0, . . . , d − 1. Then, for all n ≥ maxj{nj}, we have

HR/I(n) = d−1 X j=0 Sj(n)ζjn where Sj(n) := nj X h=0 n + αj − h − i αj− 1 ζ−jh

Observe that Sj is a polynomial in n of degree αj − 1.

Finally, we get Pi(x) := S0(x) + S1(x)ζi+ · · · + Sd−1(x)ζ(d−1)i and, by our

previous considerations, every Sj has degree less or equal to k − 2 and the

statement follows.

Example 2.18. Consider the ring R = Q[x_{1}, . . . , x4] with the graduation

given by W = [1, 2, 4, 8]. Let I = (x2

1x3− x32, x4− x41x3) ⊂ R. In this case,

we have k = 4 and d = 8.

The Hilbert quasi-polynomial PW

R = {P0, . . . , P7} is given by P0(x) = 1/384x3+ 1/16x2+ 11/24x + 1 P1(x) = 1/384x3+ 7/128x2+ 131/384x + 77/128 P2(x) = 1/384x3+ 1/16x2+ 41/96x + 7/8 P3(x) = 1/384x3+ 7/128x2+ 119/384x + 65/128 P4(x) = 1/384x3+ 1/16x2+ 11/24x + 1 P5(x) = 1/384x3+ 7/128x2+ 131/384x + 77/128 P6(x) = 1/384x3+ 1/16x2+ 41/96x + 5/8 P7(x) = 1/384x3+ 7/128x2+ 119/384x + 33/128

28 CHAPTER 2. HILBERT QUASI-POLYNOMIAL For all i = 0, . . . , 7, the polynomials Pi have degree equal to k − 1 = 3 and

the leading coefficient is 1 (k−1)!Qk

i=1di =

1 384.

Whereas the Hilbert quasi-polynomial PW

R/I = {P0, . . . , P7} of (R/I, W ) is given by P0(x) = 7/4x − 5 P1(x) = 7/4x − 27/4 P2(x) = 7/4x − 11/2 P3(x) = 7/4x − 29/4 P4(x) = 7/4x − 5 P5(x) = 7/4x − 27/4 P6(x) = 7/4x − 11/2 P7(x) = 7/4x − 29/4

and we observe that the degree of P is 1, which is less than 2 = k − 2.

### 2.3

### Proprieties of the coefficients of the Hilbert

### quasi-polynomial

In this section we present some proprieties of the coefficients of Hilbert quasi-polynomials. In particular, if we denote the Hilbert quasi-polynomial of R by PW

R = {P0, . . . , Pd−1} as usual, we show that some coefficients are

equal for all Pi, while the others change with a certain regularity, as

sug-gested by examples that we have studied. These facts could be exploited in order to reduce complexity in the computation of Hilbert quasi-polynomials.

Before showing our results, we present an example.

Example 2.19. Consider R = Q[x1, . . . , x5] with the graduation given by

W = [1, 2, 3, 4, 6]. In this case, we have k = 5 and d = 12. The Hilbert quasi-polynomial PW

2.3. PROPRIETIES OF THE COEFFICIENTS 29 P0(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 1/2x + 1 P1(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 43/108x + 1705/3456 P2(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 25/54x + 125/216 P3(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 5/12x + 75/128 P4(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 13/27x + 20/27 P5(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 41/108x + 1001/3456 P6(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 1/2x + 7/8 P7(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 43/108x + 1705/3456 P8(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 25/54x + 19/27 P9(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 5/12x + 75/128 P10(x) = 1/3456x4+ 1/108x3+ 5/48x2+ 13/27x + 133/216 P11(x) = 1/3456x4+ 1/108x3+ 19/192x2+ 41/108x + 1001/3456

We have 12 polynomials of degree 4 with leading coefficient equal to 1/3456, that’s just what we expected.

As can be seen by comparing polynomials, the following facts hold:

• The coefficient of the term of degree 3 is the same for all polynomials. • The coefficient of the term of degree 2 has periodicity 2, i.e. 5/48 is the coefficient of the term of degree 2 of all Pi(x) such that i ≡ 0 mod 2

and 19/192 is the coefficient of the term of degree 2 of all Pi(x) such

that i ≡ 1 mod 2.

• The coefficient of the term of degree 1 has periodicity 6, i.e. 1/12 is the coefficient of the term of degree 1 of all Pi(x) such that i ≡ 0 mod 6,

43/108 is the coefficient of the term of degree 1 of all Pi(x) such that

i ≡ 1 mod 6, 5/48 is the coefficient of the term of degree 1 of all Pi(x)

such that i ≡ 2 mod 6 and so on.

• The constant term seems not to have any kind of regularity.

Goal of next two results is to see if we can predict which and how coeffi-cients change.

Recall that we use δ to denote max{|I| | gcd(di)i∈I 6= 1and I ⊆ {1, . . . , k}}.

Proposition 2.20 Let (R, W ) and P_{R}W be as above. Then
P_{R}W(z) = Q(z) + R(z)

where Q(z) ∈ Q[z] is a polynomial of degree k − 1, whereas R(z) is a quasi-polynomial with rational coefficients of degree δ − 1.

30 CHAPTER 2. HILBERT QUASI-POLYNOMIAL Proof. Lemma 2.10 assures that Tδ+1 = · · · = Tk = ∅. Thus the

Hilbert-Poincaré series of (R, W ) is
HPR(t) =
Q0(t)
(1 − t)k +
X
j∈Tδ
Q(δ)_{j} (t)
(1 − ζj_{t)}δ + · · · +
X
j∈T1
Q(1)_{j} (t)
(1 − ζj_{t)} (2.12)

and, thanks to remark 2.11, we have that the coefficients of the terms of degrees δ, . . . , k − 1 don’t change and the result follows.

Let us return to the example 2.19. In that case, δ is equal to 3, in fact the
subset {2, 4, 6} is the largest subset such that its elements are not relatively
prime. By proposition 2.20, the coefficients of the terms of degree δ, . . . , k−1,
that is 3 and 4, don’t change and this is exactly what happens in the example.
Proposition 2.21Let (R, W ) and P_{R}W be as above. The rth coefficient of
P_{R}W, for r = 0, . . . , k − 1, has periodicity equal to

δr:= lcm ( gcd(di)i∈I | |I| = r + 1, I ⊆ {1, . . . , k})

Formally, the proposition asserts that if we denote the rth coefficient of Pi(t)by air, then ajr = air when j = i + δr mod d.

Proof. Let δ = max{|I| | gcd(di)i∈I 6= 1 and I ⊆ {1, . . . , k}} be as usual.

Then δr = 1for δ ≤ r ≤ k −1 and, by proposition 2.20, the assertion follows.

Let r ≤ δ − 1. Recall that Pi(x) = S0(x) + S1(x)ζi+ · · · + Sd−1(x)ζ(d−1)i,

where Sj(x) =P nj

h=0ajh

x+αj−h−1

αj−1 ζ

−jh_{. Note that each S}

j(x) doesn’t

de-pend on i and its degree is equal to αj− 1. Therefore the rth coefficient of

Pi(x) is the rth coefficient of
S0(x) + Sj1(x)ζ
i·j1 _{+ · · · + S}
jv(x)ζ
i·jv
with
{j_{1}, . . . , jv} =
[
s≥r+1
Ts =
[
r+1≤s≤δ
Ts (2.13)

where the second equality holds because of Tδ+1 = · · · = Tk = ∅.

We recall that Ts= [ I⊆{1,...,k},|I|=s gcd(di)i∈I6=1 d gcd(di)i∈I , . . . , (gcd(di)i∈I− 1) · d gcd(di)i∈I

To prove that the rth coefficient of S0(x) + Sj1(x)ζ

i·j1 _{+ · · · + S}

jv(x)ζ

i·jv

2.3. PROPRIETIES OF THE COEFFICIENTS 31 Equation (2.13).

Let N = 1, then r = δ − 1. We observe that ζδr·jt _{= 1}for all t = 1, . . . , v, (in

fact, jtcan be written as a·_{gcd(d}d_{i}_{)}_{i∈I} for some integer 1 ≤ a ≤ (gcd(di)i∈I−1),

with |I| = r + 1; since gcd(di)i∈I divides δr, then d divides δr· jt). So we

have
S0(x) + Sj1(x)ζ
i·j1_{+ · · · + S}
ju(x)ζ
i·jv _{=}
S0(x) + Sj1(x)ζ
(i+δr)·j1_{+ · · · + S}
jv(x)ζ
(i+δr)·jv

and we are done.

Let N ≥ 2, then r ≤ δ − 2, and let us suppose that the thesis is true for S

r0_{≤s≤δ}Ts, with r0 > δ − N, which means that the r0th coefficient has period

equal to δr0.

We can order the set {j1, . . . , ju, ju+1, . . . , jv}in such a way that

• j1, . . . , ju∈ Ts, with s > r + 1, and ji∈ T/ r+1, for all i = 1, . . . , u;

• ju+1, . . . , jv ∈ Tr+1.

Let γ := lcm(gcd(di)i∈I | |I| > r + 1), then by induction hypothesis we have

S0(x) + Sj1(x)ζ
i·j1_{+ · · · + S}
ju(x)ζ
i·ju _{=}
S0(x) + Sj1(x)ζ
(i+γ)·j1 _{+ · · · + S}
ju(x)ζ
(i+γ)·ju

Since γ divides δr, it follows

S0(x) + Sj1(x)ζ
i·j1_{+ · · · + S}
ju(x)ζ
i·ju _{=}
S0(x) + Sj1(x)ζ
(i+δr)·j1 _{+ · · · + S}
ju(x)ζ
(i+δr)·ju

In addition, since ζδr·jt _{= 1} for all t = u + 1, . . . , v, we have

Sju+1(x)ζ
i·ju+1 _{+ · · · + S}
jv(x)ζ
i·jv _{=}
Sju+1(x)ζ
(i+δr)·ju+1_{+ · · · + S}
jv(x)ζ
(i+δr)·jv

and then, by putting together the two equations above, we are done.

Remark 2.22. In some particular cases, for example when k >> δ or if the periods δrare quite small, knowing the period of the coefficients could be

ex-ploited to reduce the cost of computation for the Hilbert quasi-polynomials. Return to example 2.19. We already know that δ = 3, then δr = 1 for

32 CHAPTER 2. HILBERT QUASI-POLYNOMIAL For r = 2, δr is equal to 2, in fact {2, 4, 6} is the only subset of cardinality

3 such that its elements are not relatively prime and gcd(2, 4, 6) = 2. For r = 1, δr is equal to 6, in fact the subsets of cardinality 2 such that

their elements are not relatively prime are {2, 4}, {2, 6}, {3, 6} and {4, 6} and their great common divisor are respectively 2, 2, 3 and 2. Therefore, δr= lcm(2, 2, 3, 2) = 6.

### 2.4

### Formulas for the other coefficients of the Hilbert

### quasi-polynomial

In this section, we give formulas to recover the (k − 2)th and (k − 3)th coefficient of the Hilbert quasi-polynomial from the weights d1, . . . , dk, and

we illustrate how recover other coefficients.

We have already showed that if δ ≤ k − 2 (δ ≤ k − 3 respectively) then the (k − 2)th ((k − 3)th respectively) coefficient is the same for all Pi,

i = 0, . . . , d − 1. For this reason, we suppose that δ is small enough to guar-antee that the coefficient is equal for all Pi.

Proposition 2.23Let (R, W ) and P_{R}W as above. If δ ≤ k − 2, we denote
the (k − 2)th coefficient of Pi, for all i = 0, . . . , d − 1, by ck−2. Then, it holds

ck−2=

Pk

i=1di

2 · (k − 2)! ·Qk

i=1di

Proof. Let Pi(n) = S0(n) + S1(n)ζn+ · · · + Sd−1(n)ζn·(d−1) as usual, where

Sj(n) = nj X h=0 ajh n + αj− h − 1 αj− 1 ζ−jh.

Since δ ≤ k − 2 and by remark 2.11, the (k − 2)th coefficient of Pi is given

by the (k − 2)th coefficient of S0(n). Let ai:= a0i, then we have

S0(n) = a0 n + k − 1 k − 1 +a1 n + k − 2 k − 1 +· · ·+ak−1 n k − 1 +ak n − 1 k − 1 =

2.4. FORMULAS FOR THE OTHER COEFFICIENTS 33 = a0· (n + k − 1) · · · (n + 1) (k − 1)! + a1· (n + k − 2) · · · (n) (k − 1)! + + · · · + ak−1· (n) · · · (n − k + 2) (k − 1)! + ak· (n − 1) · · · (n − k + 1) (k − 1)! = k−1 X i=0 cini (2.14) By this equation, we get

ck−2= a0 (k − 1) + (k − 2) + · · · + 1 (k − 1)! + a1 (k − 2) + (k − 3) + · · · + 1 (k − 1)! + + · · · + ak (−1) + (−2) + · · · + (−k + 1) (k − 1)! = = a0 (k − 1)!· k(k − 1) 2 + a1 (k − 1)!· (k − 1)(k − 2) 2 +· · ·+ ak (k − 1)!· (−k)(k − 1) 2 = = 1

2(k − 2)![ka0+ (k − 2)a1+ · · · + (k − 2k)ak] =

= 1 2(k − 2)! " k k X i=0 ai− 2 k X i=0 (i · ai) # (2.15) We observe that k X i=0 ai= Q0(1) and k X i=0 i · ai = Q00(1) (2.16) Since Q0(t) = A1(1 − t)k−1+ A2(1 − t)k−2 + · · · + Ak−1(1 − t) + Ak, by

evaluating Q0(t)and Q00(t)at t = 1 and by putting together Equation (2.15)

and Equation (2.16), we have ck−2= 1 2(k − 2)!k · Q0(1) − 2 · Q 0 0(1) = 1 2(k − 2)![k · Ak+ 2 · Ak−1] . By the following lemma and by recalling that Ak = Qk1

i=1di, the proposition

is proved.

Lemma 2.24 The constant Ak−1 which appears in Equation (2.6) of the

expression of Q0(t)is such that

Ak−1 =

Pk

i=1di− k

2 ·Qk

34 CHAPTER 2. HILBERT QUASI-POLYNOMIAL

Proof. In order to obtain an expression for A_{k−1}, evaluating at t = 1 the
derivative of the Equation (2.7) is a good strategy. Before doing it, we
observe that the evaluation at t = 1 of the derivative of the equation gives
0 except for the terms in the first summation for t = k − 1 and t = k. For
this reason, we’ll consider only these addends and we’ll calculate only their
derivatives.
0 = ∂
∂t
(Ak−1(1 − t) + Ak) ·
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
0 = −Ak−1·
Y
j∈Tδ
(1 − ζj)δ· · · Y
j∈T1
(1 − ζj)+
+ Ak·
∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1

We already know that Ak = 1 Qk i=1di and Y j∈Tδ (1 − ζj)δ· · · Y j∈T1 (1 − ζj) = k Y i=1 di. So, we have Ak−1 = 1 (Qk i=1di)2 · ∂ ∂t Y j∈Tδ (1 − ζjt)δ· · · Y j∈T1 (1 − ζjt) t =1 (2.17)

and it remains to calculate ∂
∂t
h
Q
j∈Tδ(1 − ζ
j_{t)}δ_{· · ·}Q
j∈T1(1 − ζ
j_{t)}i and
eval-uate it at t = 1.

Consider the following equations g(t) (1 − t)k = Y j∈Tδ (1 − ζjt)δ · · · Y j∈T1 (1 − ζjt) g(t) (1 − t)k = k Y j=1 1 − tdj 1 − t = k Y j=1 1 + t + · · · + tdj−1

2.4. FORMULAS FOR THE OTHER COEFFICIENTS 35
Then, it holds
∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
=
∂
∂t
k
Y
j=1
1 + t + · · · + tdj−1
=
=
k
X
i=1
Y
j6=1
1 + t + · · · + tdj−1
1 + 2t + · · · + (di− 1)tdi−2
Finally, we have
∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
=
k
X
i=1
Y
j6=i
dj
di(di− 1)
2
=
=
k
Y
i=1
di·
k
X
i=1
di− 1
2
!
= 1
2·
k
Y
i=1
di·
" _{k}
X
i=1
di− k
#

and by substituting this expression in Equation (2.17), we are done.

We are now interested in giving a similar formula for the (k − 3)th coef-ficient of PW

R .

Proposition 2.25Let (R, W ) and PW

R as above. If δ ≤ k − 3, we denote

the (k − 3)th coefficient of Pi, for all i = 0, . . . , d − 1, by ck−3. Then, it holds

ck−3 = 3Pk i=1di 2 −Pk i=1d2i 24(k − 3)!Qk i=1di

Proof. Given i = 0, . . . , d−1, let Pi(n)and S0(n)be as usual. Since δ < k−3

and by remark 2.11, we have that the (k−3)th coefficient of Pi(n)correspond

36 CHAPTER 2. HILBERT QUASI-POLYNOMIAL ck−3 = a0 2(k − 1)!· (k − 1) · k−1 X j=1 j − (k − 1) + +(k − 2) · k−1 X j=1 j − (k − 2) + · · · + 1 · k−1 X j=1 j − (1) + + a1 2(k − 1)!· (k − 2) · k−2 X j=0 j − (k − 2) + +(k − 3) · k−2 X j=0 j − (k − 3) + · · · + 1 · k−2 X j=0 j − (1) + + ... + + ak 2(k − 1)!· (−1) · −1 X j=−k+1 j − (−1) + (−2) · −1 X j=−k+1 j − (−2) + · · · + (−k + 1) · −1 X j=−k+1 j − (−k + 1) = a0 2(k − 1)!· k(k − 1) 2 2 − k−1 X j=1 j2 + + a1 2(k − 1)!· (k − 1)(k − 2) 2 2 − k−2 X j=0 j2 + + ... + + ak 2(k − 1)!· (k − 1)(−k) 2 2 − −1 X j=−k+1 j2 = k X i=0 ai 2(k − 1)! (k − 1)(k − 2i) 2 2 − k−i−1 X j=−i+1 j2

2.4. FORMULAS FOR THE OTHER COEFFICIENTS 37 We observe that for all i = 0, . . . , k it holds

ai 2(k − 1)! (k − 1)(k − 2i) 2 2 − k−i−1 X j=−i+1 j2 = ai 24(k − 1)! ·3k

4_{+ (−12i − 10)k}3_{+ (12i}2_{+ 36i + 9)k}2

+ (−36i2− 24i − 2)k + 24i2]

thus we have
ck−3 =
1
24(k − 1)!·
"
(3k4− 10k3_{+ 9k}2_{− 2k) ·}
k
X
i=0
ai+
+ (−12k3+ 36k2− 24k) ·
k
X
i=0
iai+
+ (12k2− 36k + 24) ·
k
X
i=0
i2ai
#
(2.18)
We recall that
k
X
i=1
ai = Q0(1) and
k
X
i=1
iai = Q00(1)

and we observe that

k

X

i=1

i2ai = Q000(1) + Q0(1)

By substituting these expressions in Equation 2.19, we obtain
ck−3 =
1
24(k − 1)!· [ (3k
4_{− 10k}3_{+ 9k}2_{− 2k) · Q}
0(1)+
+(−12k3+ 36k2− 24k) · Q0_{0}(1)+
+(12k2− 36k + 24) · (Q00_{0}(1) + Q0_{0}(1))
= 1
24(k − 1)!· [ (3k
4_{− 10k}3_{+ 9k}2_{− 2k) · Q}
0(1)+
+(−12k3+ 48k2− 60k + 24) · Q0_{0}(1)+
+(12k2− 36k + 24) · Q00_{0}(1) ] (2.19)
We have already seen that

Q0(1) = 1 Qk i=1di and Q0 0(1) = − Pk i=1di− k 2Qk i=1di

38 CHAPTER 2. HILBERT QUASI-POLYNOMIAL Since Q0(t) = A1(1−t)k−1+A2(1−t)k−2+· · ·+Ak−1(1−t)+Ak, after having

calculated Q00

0(t) and having evaluated it at t = 1, we get Q000(1) = 2Ak−2.

Therefore, it remains to calculate Ak−2. We’ll show in Lemma 2.26 that

Ak−2= 1 24Qk i=1di 3 k X i=1 di− k !2 − k X i=1 (di− 1)(di− 5)

For now let us suppose it holds, and we get
Q00_{0}(1) = 1
12Qk
i=1di
3
k
X
i=1
di− k
!2
−
k
X
i=1
(di− 1)(di− 5)

By substituting the expressions of Q0(1), Q00(1) and Q000(1) in Equation

(2.19), we have ck−3 = 1 24(k − 1)!Qk i=1di · { (3k4− 10k3+ 9k2− 2k)+ +(6k3− 24k2+ 30k − 12) · ( k X i=1 di− k)+ + (k2− 3k + 2) · 3 k X i=1 di− k !2 − k X i=1 (di− 1)(di− 5) = 1 24(k − 3)!Qk i=1di · " k(3k − 1) + 6(k − 1) · k X i=1 di− k ! + + 3 k X i=1 di− k !2 − k X i=1 (di− 1)(di− 5)

After having made calculations, we get ck−3 = 3 ·Pk i=1di 2 −Pk i=1d2i 24 · (k − 3)! ·Qk i=1di

and then we are done.

Lemma 2.26The constant A_{k−2} which appears in Equation (2.6) of the
expression of Q0(t)is such that

Ak−2 = 1 24 ·Qk i=1di 3 k X i=1 di− k !2 − k X i=1 (di− 1)(di− 5)

2.4. FORMULAS FOR THE OTHER COEFFICIENTS 39 Proof. In order to obtain an expression for Ak−2, we’ll act in a similar way

to which it is done in the calculus of Ak−1. So we are going to evaluate at

t = 1 the second derivative of the Equation (2.7). But before doing it, we
observe that the evaluation at t = 1 of the second derivative of the equation
gives 0 except for the terms in the first summation for k − 2 ≤ t ≤ k. For
this reason, we’ll consider only these addends and we’ll calculate only their
derivatives.
0 = ∂
2
∂t2
k
X
t=k−2
(1 − t)k−t· Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
0 = ∂
∂t
(−2Ak−2(1 − t) − Ak−1) ·
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)+
+
k
X
t=k−2
(1 − t)k−t· ∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
0 =
2Ak−2
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)+
+ (−4Ak−2− 2Ak−1) ·
∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
+
+
k
X
t=k−2
(1 − t)k−t· ∂
2
∂t2
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
0 = 2Ak−2·
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
+
− 2A_{k−1}· ∂
∂t
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
+
+ Ak·
∂2
∂t2
Y
j∈Tδ
(1 − ζjt)δ· · · Y
j∈T1
(1 − ζjt)
t =1
(2.20)