Keywords: Nombres de Betti, Fonction de Hilbert non standard, Fonction de partition vectorielle Betti numbers, Nonstandard Hilbert function, Vector partition function

Fonction de Hilbert non standard et nombres de Betti gradués des puissances d'idéaux

En utilisant le concept des fonctions de partition , nous étudions le comportement asymptotique des nombres de Betti gradués des puissances d'idéaux homogènes dans un polynôme sur un corp.

.

La déclaration principale traite le cas des produits des puissances d'idéaux homogènes dans un algèbre Z d -graduée , pour un graduer positif, dans le sens de [37] et il est généralise également pour les filtrations I -good .

Dans la deuxième partie, en utilisant la version paramétrique de l'algorithme de Barvinok, nous donnons une formule fermée pour les fonctions de Hilbert non-standard d'anneaux de polynômes, en petites dimensions.

acknowledgments

Foremost, I would like to express my sincere gratitude to my advisor Prof. Marc CHARDIN for the continuous support of my Ph.D study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I am proud that I passed my Master thesis under supervision of Prof. Marc CHARDIN , I should say that my graduate knowledge is indebted to Prof. Marc and I will appreciate him in all my life. I would like to thank for all of the love to my wife Mozhgan for her personal support and great patience at all times,as always, for which my mere expression of thanks likewise does not suffice. My parents, brother have given me their unequivocal support throughout . I am using this opportunity to express my gratitude to Prof.Siamak Yassemi I was his student during my B.Sc degree at University of Tehran, where I realy motivated in commutative algebra and I benefited from his advice during his several research visitings in Paris .

I would like to acknowledge the academic and financial support of University of Pierre et Marie Curie.

Many thanks to all of my friends in Paris, specially Shahab and Maziar.

INTRODUCTION

The study of homological invariants of powers of ideals goes back, at least, to the work of Brodmann in the 70's and attracted a lot of attention these last two decades.

One of the most important results in this area is the result on the asymptotic linearity of Castelnuovo-Mumford regularity obtained by Kodiyalam [START_REF] Kodiyalam | Asymptotic behaviour of Castelnuovo-Mumford regularity[END_REF] and Cutkosky, Herzog and Trung [START_REF] Cutkosky | Asymptotic behaviour of the Castelnuovo-Mumford regularity[END_REF], independently.

The proof of Cutkosky, Herzog and Trung further shows the eventual linearity in t of end Tor S i (I t , k) := max{µ| Tor S i (I t , k) µ = 0}.

It is natural to investigate the asymptotic behavior of Betti numbers β i (I t ) := dim k Tor S i (I t , k) as t varies. In [START_REF] Northcott | Reductions of ideals in local rings[END_REF], Northcott and Rees already investigated the asymptotic behavior of β k 1 (I t ). Later, using the Hilbert-Serre theorem, Kodiyalam [START_REF] Kodiyalam | Homological invariants of powers of an ideal[END_REF]Theorem 1] proved that for any non-negative integer i and sufficiently large t , the i -th Betti number, β k i (I t ), is a polynomial Q i in t of degree at most the analytic spread of I minus one.

Recently, refining the result of [START_REF] Cutkosky | Asymptotic behaviour of the Castelnuovo-Mumford regularity[END_REF] on end Tor S i (I t , k) , [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] gives a precise picture of the set of degrees γ such that Tor S i (I t , A) γ = 0 when t varies in N. The article [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] considers a polynomial ring S = A[x 1 ,..., x n ] graded by a finitely generated abelian group G over a Noetherian ring A, see [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF]Theorem 4.6].

When A = k is a field and the ideal is generated by a single degree d ∈ G, it is proved that for any γ and any j , the function dim k Tor S i (I t , k) γ+td is a polynomial in t for t 0, See [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF]Theorem 3.3] and [START_REF] Singla | Onvex-geometric, homological and combinatorial properties of graded ideals[END_REF].

My Thesis is here interested in the behavior of dim k Tor S i (I t , k) γ when I is an arbitrary graded ideal and S = k[x 1 ,..., x n ] is a Z p -graded polynomial ring over a field k, for a positive grading in the sense of [START_REF] Miller | Combinatorial commutative algebra[END_REF].

In the case of a positive Z-grading, the result takes the following form:

Theorem 1.0.1. (See Theorem 5. 1.2). Let S = k[x 1 ,..., x n ] be a positively graded polynomial ring over a field k and let I be a homogeneous ideal in S.

There exist, t 0 , m, D ∈ Z, linear functions L i (t ) = a i t + b i , for i = 0,...,m, with a i among the degrees of the minimal generators of I and b i ∈ Z, and polynomials Q i , j ∈ Q[x, y] for i = 1,...,m and j ∈ 1,...,D, such that, for t ≥ t 0 , (i) L i (t ) < L j (t ) ⇔ i < j , (ii) If µ < L 0 (t ) or µ > L m (t ), then Tor S i (I t , k) µ = 0.

(iii) If L i -1 (t ) ≤ µ ≤ L i (t ) and a i tµ ≡ j mod (D), then

dim k Tor S i (I t , k) µ = Q i , j (µ, t ).
Our general result, Theorem 4.2.4, involves a finitely generated graded module M and a finite collection of graded ideals I 1 ,..., I s . The grading is a positive Z p -grading, and a special type of finite decomposition of Z p+s is described in such a way that in each region dim k (Tor S i (M I t 1 1 ...I t s s , k) γ ) is a quasipolynomial in (γ, t 1 ,..., t s ), with respect to a lattice defined in terms of the degrees of generators of the ideals.

The central object in this study is the Rees modification M R I . This graded object admits a graded free resolution over a polynomial extension of S, from which we deduce a Z p+s -grading on Tor modules as in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF]. Investigating Hilbert series of modules for such a grading, using vector partition functions, leads to the results.

In the last chapter we will study the structure and dimension of each pieces of Tor module of I -good filtrations the main result takes the following form : (1)Then for all i , j :

Tor A i (Tor R j (R ϕ , A), k
) is finitely generated k[T 1 ,...,T r ]-module .

(2) There exist, t 0 , m, D ∈ Z, linear functions L i (t ) = a i t + b i , for i = 0,...,m, with a i among the degrees of the minimal generators of I and b i ∈ Z, and polynomials Q i , j ∈ Q[x, y] for i = 1,...,m and j ∈ 1,...,D, such that, for t ≥ t 0 , (i) L i (t ) < L j (t ) ⇔ i < j , (ii) If µ < L 0 (t ) or µ > L m (t ), then Tor S i (ϕ(t ), k) µ = 0.

(iii) If L i -1 (t ) ≤ µ ≤ L i (t ) and a i tµ ≡ j mod (D), then

dim k Tor S i (ϕ(t ), k) µ = Q i , j (µ, t ).
Other interest of my thesis is about Non-standard Hilbert functions, actually non-standard Hilbert functions first raised in the in Gabber's proof of Serre non-negativity conjecture [START_REF] Roberts | Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture[END_REF]. It has been studied by several authors [START_REF] Hoang | Hilbert polynomials of non-standard bigraded algebras[END_REF][START_REF] Roberts | Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture[END_REF][START_REF] Roberts | Intersection multiplicities and Hilbert polynomials[END_REF]. As it was noticed in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] , the module ⊕ t Tor S i (I t , k) for a homogeneous ideal I in graded ring S has the structure of a finitely generated graded module over a non-standard graded polynomial ring over k , from which one can deduce the behavior of Tor S i (I t , k) when t varies.

It is also desirable to give closed formula for quasi-polynomials coming from a vector partition function. However, in general, such a formula doesn't exist . An algorithm that uses a continued fraction expansion and gives closed formula for the generating function corresponding to a two dimensional polytope was given by Barvinok. We use a parametric version of his algorithm and deduce the Hilbert

function of polynomial ring B = k[T 1 ,...,T n ] such that {deg T i |1 i n} = {(d j , 1)|1 j 4} more pre- cisely for the associated polytope P (b = (b 1 , b 2 )) = {x ∈ R r |Ax = b; x 0}, where A =    d 1 d 2 d 3 d 4 1 1 1 1   ,
   d 1 d 2 d 3 d 4 1 1 1 1    for d 1 < d 2 < d 3 < d 4
then there is a one to one correspondence between the integer points of P (b) and Q ⊂ R 2 and we have the following about Q:

1. Q = (λ 1 , λ 2 ) ∈ R 2 |λ 1 (d 2 -d 1 ) 0; λ 2 (d 2 -d 1 ) 0; λ 1 (d 1 -d 4 ) + λ 2 (d 1 -d 3 ) + d 1 b 2 -b 1 d 2 -d 1 0; λ 1 (d 4 -d 2 ) + λ 2 (d 3 -d 2 ) + b 1 -d 2 b 2 d 2 -d 1 0 f or b 1 , b 2 > 0 ,
2. Q has the following vertices :

Q 1 = ( d 3 b 2 -b 1 (d 2 -d 1 )(d 4 -d 3 ) ,( b 1 -d 4 b 2 (d 2 -d 1 )(d 4 -d 3 ) ) , Q 2 = ( d 2 b 2 -b 1 (d 2 -d 1 )(d 4 -d 2 ) , 0) , Q 3 = (0, d 2 b 2 -b 1 (d 2 -d 1 )(d 3 -d 2 ) ) , Q 4 = ( b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 4 ) , 0) , Q 5 = (0, b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 3 ) ) , Q 6 = (0, 0), 3. The generation function of Q in the first chamber C 1 is : f C 1 (Q, x) = 1 (1-x -1 1 )(1-x -1 2 ) + x s 1 1 (1-x 1 )(1-x -1 2 ) - x s 1 1 x -( a 0 s 1 +a 0 s 1 ) 2 (1-x -1 2 )(1-x 1 x -a 0 2 ) + x ( (a 0 a 1 +1)s 1 -a 1 a 0 s 1 ) 1 x -(a 0 (a 0 a 1 +1)s 1 -(a 0 a 1 +1) s 1 ) 2 (1-x 1 x -a 0 2 )(1-x a 1 1 x -(a 1 a 0 +1) 2 ) + x -s 2 2 (1-x -1 2 )(1-x 1 x -a 0 2 ) - x ( -a 1 s 2 +a 1 s 2 ) 1 x -(a 0 -a 1 s 2 +(a 0 a 1 +1) s 2 ) 2 (1-x 1 x -a 0 2 )(1-x a 1 1 x -(a 1 a 0 +1) 2 )
Where s

1 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 4 ) and s 2 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 3 )
.

Finally evaluating f C 1 (Q, x) at x = (1, 1) gives us HF (B,(b 1 , b 2 )
). But our generating function has a pole at x = (1, 1), so that we use the Yoshida et al [START_REF] De Loera | Effective lattice point counting in rational convex polytopes[END_REF] method to find the explicit formula of the Hilbert function of B from theorem 3.5.6 .

C H A P T E R

) R = R 0 ⊕ R 1 ⊕ . . . such that R m .R n ⊆ R m+n for all m, n ∈ Z ≥0
, and where R 0 is a subring (i.e. 1 ∈ R 0 ). A Z-graded ring is one where the decomposition is into R = ⊕ n∈Z R n . In either case, the elements of the subgroup R n are called homogeneous of degree n.

Let R be a ring and x 1 ,..., x n indeterminates over R.

For u = (u 1 , ••• , u n ) ∈ N n let x u = x u 1 1 ••• x u n n , then
one can consider the polynomial ring S = R[x 1 ,..., x n ] as a graded ring by the total degree of polynomial 9

where in this case the graded pieces can be write as

S m = { m u i ∈N n ;i =1 a u i x u i |a u i ∈ R and u 1 + ••• + u n = m} Definition 2.
i = n j for i = j .
In the category of graded R-modules, the morphisms of R-modules are the ones that preserve the grading. In other words, morphisms of graded modules take homogeneous elements to homogeneous elements of the same degree.

Definition 2.1.3. If M is a graded module, we set M (n) for the same R-module but with the grading

M (n) k = M n+k .
A graded homomorphism (of degree 0 )between graded R-modules M , N is defined to be an R-module homomorphism sending M n to N n for any n.

Example 2.1.4.

1. If R is a graded ring, then R is a graded module over itself.

2. If S is any ring, then S can be considered as a graded ring with S 0 = S and S i = 0 for i > 0. Then a graded S-module is just a Z-indexed collection of (ordinary) S-modules.

3. (The blow-up algebra ,also called Rees algebra) Let S be any ring, and let J be an ideal of S. We can make R = S ⊕ J ⊕ J 2 ⊕ . . . (the blow-up algebra) into a graded ring, by defining the multiplication from the one of S after noticing that J i J j ⊆ J i + j . Given any S-module M , there is a graded R- (1) N is a graded R-module.

module M ⊕ J M ⊕ J 2 M ⊕ . . .,
(2) N = n N M n .

(3

) If u = u 1 + ••• + u n ∈ N then u i ∈ N for 1 i n.
(4) N has a homogeneous set of generators.

Rees filtration

The notion of Rees algebra is classically extended as follows :

Definition 2.1.7. Let R be a commutative ring and

I = {I n } ∞ n=0 a sequence of ideals of R. Then I is called filtration of R if, ( 1
) I 0 = R (2) I n ⊃ I n+1 ∀n ∈ N (3) I n .I m ⊆ I n+m .
Let I be a filtration of R then we define the Rees algebra and associated graded ring associated to I by

R I = ⊕ n 0 I n , g r I = ⊕ n 0 I n I n+1
One of the most impotent example is when a filtration I is given by a power of ideal I , in this case if

I is generated by { f 1 , ••• , f m } ⊂ R
ψ : R [T 1 , •••T n ] → ψR [I t] , ψ(T i ) = f i t Remark 2.1.8.
If R is a ring with a filtration I given by powers of an ideal I , then R[I t] is a Noetherian.

If a Rees ring with respect to a filtration I is Noetherian, then I is called a Noetherian filtration . The following proposition gives some equivalent conditions. Proposition 2.1.9. Let R be a ring with a filtrationI = {I n } ∞ n=0 . The following conditions are equivalent:

(i) I is Noetherian.

(ii) R is Noetherian, and R I is finitely generated over R.

(iii) R is Noetherian, and R I + is finitely generated over R.

Graded Free Resolution

Definition 2.1.10. Let M be a finite module over local ring (S, m, k) , the following exact sequence is called a minimal free resolution

•••F n -→ d n F n-1 -→ d n-1 • • • -→ F 0 -→ d -→ M -→ 0
if it statisfies the following conditions :

(1) F i are free S-module for all i ∈ N

(2)

d i (F i ) ⊂ mF i -1 (3)d : F 0 k -→ M k is an isomorphism
The Hilbert Syzygy theorem says that every module M over a polynomial ring S over a field has a free resolution with length at most the number of variables.

0 → F s → ... → F 1 → F 0 → M → 0
with s n and the rank of the F i 's in any minimal resolution only depends on M .

We can choose all the generators of various syzygy modules to be homogeneous and we can define the generators of the free modules in a way that all the maps are of degree zero.

Furthermore, in each step, if we choose a minimal generating set for the syzygy modules, we get a minimal free resolution of M . In this way, we can write

F i = j ∈Z S(-j ) β i , j
for some natural numbers β i , j . These numbers form a set of invariants of M as a graded S-module and we can also obtain them as the homological invariants β i , j (M ) = dim k Tor S i (M , k) j . These numbers are called graded Betti numbers.

Indeed, a minimal free resolution is an invariant associated to a graded module over a ring graded by the natural numbers N, or more generally by N n . The information provided by free resolutions is a refinement of the information provided by the Hilbert polynomial and Hilbert function.

Proposition 2.1.12. If F : ...F 1 → F 0 is the minimal free resolution of a finitely generated graded S-module M , and k denotes the residue field S/m, then any minimal set of homogeneous generators of F i contains precisely dim k Tor S i (k, M ) j generators of degree j.

Proposition 2.1.13. Let {β i , j } be the graded Betti numbers of a finitely generated S-module. If β i , j = 0 for all j ≤ d then β i +1, j = 0 for all j ≤ d.

Hilbert functions and Hibert seies

We can define graded modules similarly to the classical Z-graded case. When G = Z d and the grading is positive, (generalized) Laurent series are associated to finitely generated graded modules: Definition 2.1.14. The Hilbert function of a finitely generated module M over a positively graded polynomial ring is the map:

HF (M ; -) : Z d -→ N µ -→ dim k (M µ ).
The Hilbert series of M is the Laurent series:

H (M ; t ) = µ∈Z d dim k (M µ )t
H (S(-µ); t ) = t µ n i =1 (1 -t µ i )
.

where

µ i = deg(x i ).
2. If M is a finitely generated graded S-module, setting

Σ M := ∪ Supp Z d (Tor R (M , k)) and κ M (t ) := a∈Σ M (-1) dim k (Tor R (M , k)) a t a ,
one has H (M ; t ) = κ M (t )H (S; t ).

Closures of ideals

Here we recall some basic fact of closures about ideal froms [START_REF] Vasconcelos | Integral closure[END_REF] which is usefull for the last chapter.

Integral Closure

Definition 2.1.17. Let R ⊆ S be rings. An element f ∈ S is integral over R if f satisfies a monic polynomial equation

f n + a 1 f n-1 + ... + a n = 0
with coefficients in R. The integral closure of R in S is the set of all elements of S integral over R, it turns out to be a subring of S.

The ring R is integrally closed in S if all elements of S that are integral over R actually belong to R. The ring R is normal if it is integrally closed in the ring obtained from R by inverting all non-zerodivisors.

CHAPTER 2. PRELIMINARIES

Instead of a ring, the integral closure of an ideal is defined as follows:

Definition 2.1.18. Let R be a ring and I be an ideal of R. An element z ∈ R is integral over I if it satisfies the following equation.

z n + a 1 z n-1 + ••• + a n = 0 a i ∈ I i
The set of all integral elements over I is called integral closure of I and denoted by I Suppose

that H 1 R+ (R) 0 = 0 . Then a 1 R+ (R) + 1 = ρ(I ). Where R = ⊕ n 0 I n and a 1 R+ = sup{n ∈ Z| H 1 R+ (R) n = 0}.

Spectral sequence

In this section, we are going to collect some necessary notations and terminologies about spectral sequences from [START_REF] Weibel | An introduction to homological algebra[END_REF].

Definition 2.1.22. A spectral sequence in an abelian category A is a collection of the following data:

1. A family {E r p,q } defined for all p, q ∈ Z.

2. Maps d r p,q : E r p,q → E r p-r,q+r -1 are differentials in the sense that d r d r = 0.

3. There is isomorphisms between E r +1 p,q and homology of E r * , * at the spot E r p,q :

E r +1 p,q ∼ = ker(d r p,q ) Im(d r p+r,q-r +1 ) Definition 2.1.23. 1.
A spectral sequence is called bounded if for each n there are only finitely many nonzero terms of total degree n in E n * , * , more precisely there is an n 0 such that E n p,q = E n+1 p,q for all p, q ∈ Z and n n 0 . We represent the stable value of E n p,q by E ∞ p,q .

2. A bounded spectral sequence is called converges to a given family {H n } of objects of an abelian category A, if we have a following filtration for each H n :

0 = F s H n ⊆ ••• ⊆ F p-1 H n ⊆ F p H n ⊆ F p+1 H n ⊆ •••F t H n = H n
and we are given isomorphisms :

E ∞ p,q = F p H p+q F p-1 H p+q
and we write E r p,q ⇒ H p+q to represent this fact .

Spectral Sequence of Double Complex

Let C = C * , * be a double complex in an abelian category A with total complex Tot (C ) n = ⊕ i + j =n C i , j . We define two filtrations on the total complex by rows and columns.

.

.. ... ... ... C p-1,q+1 o o C p,q+1 o o C p+1,q+1 o o ... o o ... C p-1,q o o C p,q o o C p+1,q o o ... o o ... C p-1,q-1 o o C p,q-1 o o C p+1,q-1 o o ... o o ... ... ...
More precisely we define two filtrations I F p (Tot (C )) and I I F p (Tot (C )) as follows:

I F n p (Tot (C )) = m p C m,n-m I I F n p (Tot (C )) = m p C n-m,m
Theorem 2.1.24. Let C be a first quadrant double complex in an abelian category A. There are two spectral sequences E n and E n determined by I F p (Tot (C )) and I I F p (Tot (C )) filtrations respectively with

E 0 p,q = C p,q , E 1 p,q = H q ver t (C p, * ) , E 2 p,q = H p hor H q ver t (C ) , E 0 p,q = C q,p , E 1 p,q = H q hor (C * ,p ) , E 2 p,q = H p ver H q hor (C )
Both spectral sequences E n and E n converge to H p+q (Tot (C )).

One of the applications of these different spectral sequences with the same convergence could be for computing Tor modules, more precisely here we recall the Base-change theorem about Tor modules which is useful in the last chapter .

Theorem 2.1.25. Let f : R → S be a ring map. Then there is a first quadrant homology spectral sequence

E 2 p,q = Tor S p (Tor R q (A, S), B ) ⇒ Tor R p+q (A, B )
for every R-module A and S-module B.

Castelnuovo-Mumford Regularity

One of the most important invariants which measures the complexity of a coherent sheaf F , on P r is the Castelnuovo-Mumford regularity. It was first introduced by Mumford in [START_REF] Mumford | Lectures on curves on an algebraic surface[END_REF]Chapter 14] as how much one has to twist a coherent sheaf F in order for the higher cohomology to vanish. Alternatively a

coherent sheaf F is called m-regular if, H i (P r , F ⊗ P r (m -i )) = 0 for all i > 0 then reg(F ) = min{m ∈ Z|F is m -regular}.
A related idea in commutative algebra was given by Eisenbud and Goto [START_REF] Eisenbud | Linear free resolutions and minimal multiplicity[END_REF].

Let R = k [x 1 , ••• , x n ] be
a polynomial ring over a field k and M a finitely generated graded R-module. Assume M has a minimal graded free resolution as:

F • : 0 → F t → ... → F 2 → F 1 → F 0 → M → 0 Set a i (M ) := max{µ|H i m (M ) µ = 0} if H i m (M ) = 0 and a i (M ) := ∞ else, b i (M ) := max{µ|Tor R i (M , k) µ = 0} if Tor R i (M , k) = 0 and b i (M ) := ∞ else, then the Castelnuovo-Mumford regularity of M is defined as: r eg (M ) := max i {a i (M ) + i } = max i {b i (M ) -i }.
Central motivation of current thesis concerns the remarkable result about linearity behavior of regularity of powers of graded ideal I in R due to Kodiyalam [START_REF] Kodiyalam | Asymptotic behaviour of Castelnuovo-Mumford regularity[END_REF], Coutkosky,Herzog and Trung [START_REF] Cutkosky | Asymptotic behaviour of the Castelnuovo-Mumford regularity[END_REF] , Trung and

Wang [START_REF] Trung | On the asymptotic behavior of Castelnuovo-Mumford regularity[END_REF] and Chardin [START_REF] Chardin | Powers of ideals and the cohomology of stalks and fibers of morphisms[END_REF] . We state it as:

Theorem 2.1.26. Let A be a standard graded Noetherian algebra . If I is a graded ideal and M = 0 a finitely generated graded A-module, then there exists n 1 and b such that

reg(I n M ) = nd + b, ∀n n 1 with d := min{µ| ∃m 1, (I µ )I m-1 M = I m M }
Similar question studied for ideal sheaves, but the behavior is much more complicated than for graded ideals. Let I be a graded ideal generated in degree at most d in the standard algebra over a field and m = A + . We denote the saturation of I with respect to m by I sat := n (I : m n ) , it has been shown in [START_REF] Cutkosky | Asymptotic behaviour of the Castelnuovo-Mumford regularity[END_REF] that the regularity of (I n ) sat is in several cases not eventually linear but at least Cutkosky, Ein and Lazarsfeld proved in [] that the following limit exists

s(I ) = lim n→∞ reg(I n ) sat n
where s(I ) equals the invesrse of a Seshadri constant.

Discrete Geometry

Polytopes

There are two notions of polytopes (H-polytope and V-polytope ) where their equivalence have been proved in the main theorem of polytopes (see 2.2.3) but first we look at their definitions and related concepts .

Definition 2.2.1.

1. A hyperplane in R d is a set H := {x ∈ R d |a 1 x 1 + ••• + a d x d = b}. 2. A convex combination of finite points q 1 , ••• , q t is a set com({q 1 , ••• , q t }) := {x ∈ R d | t i =1 λ i q i f or λ i 0 , λ i = 1}
The set of all convex combination of the points

q 1 , ••• , q t is called convex hull of q 1 , ••• , q t and it denoted by conv(q 1 , ••• , q t ).
3. The minkowski sum of two sets P,Q ⊆ R d is defined as:

P + Q = {x + y|x ∈ P, y ∈ Q} Definition 2.2.2.
1. A H-polyhedron in R d denotes as the intersection of closed halfspaces in R d in the following form

P = {x ∈ R d |Ax b} where A ∈ R (n×d ) and b ∈ R n . A bounded H-polyhedron is called H-polytope.
2. A V-polyhedron in R d denotes as the convex hull of a finite number of points and it is of form

P = conv(q 1 , ••• , q t ) . A bounded V-polyhedron is called V-polytope.

Theorem 2.2.3. [53] Every V-polytope has a description by inequalities as H-polytope and every H-polytope

is the convex hull of minimal number of finitely many points .

Based on the above theorem, we will only use the term polytope for both notions of polytops.

Faces of Polytopes

Definition 2.2.4. Let P ⊆ R d be convex polytope. The inequality c.x c 0 is called valid for P if all points

x ∈ P satisfy this inequality. The hyperplane

H = {x ∈ R d |c.x = c 0 } is called supporting hyperplane of P . A
face of P is set of form F = P ∩ H and the dimension of a face of P is the dimension of its affine hull.

Gale Diagrams

Let

n points v 1 , ••• , v n be in R d -1
whose affine hull has dimension d -1 . Consider the matrix

A :=    1 1 ••• 1 v 1 v 2 ••• v n   
the kernel of A is defined as:

ker(A) := {x ∈ R n |Ax = 0} Definition 2.2.5. Let x ∈ R n and v 1 , ••• , v n vectors in R n . (1) If n i =1 v i xi = 0 and n i =1 x i = 0 then x is called an affine dependence relation on v 1 , ••• , v n . (2) If n i =1 v i xi = 0 then x is called a linear dependence relation on v 1 , ••• , v n . (3) If x = 0 is the only solution of n i =1 v i xi = 0 and n i =1 x i = 0 then v 1 , ••• , v n is called affinely indepen- dent. Let B 1 , ••• , B n-d ∈
R n be a basis for the vector space ker(A) and put them as the columns of n × (nd ) matrix.

B := B 1 B 2 ••• B n-d Definition 2.2.6. Let G ={b 1 , ••• , b n } ⊂ R n-d be the n ordered rows of B . Then G is called a Gale transform of {v 1 , ••• , v n } . The Gale diagrams of {v 1 , ••• , v n } is the vector configuration G in R n-d . Example 2.2.7. Let A =    d 1 ... d n 1 ... 1    be a 2 × n-matrix with entries in N such that d 1 ≤ ... ≤ d n .
By computing a basis for the kernel of A, we have:

B =                           d 2 -d 3 d 2 -d 4 ••• d 2 -d n d 3 -d 1 d 4 -d 1 ••• d n -d 1 d 1 -d 2 0 ••• 0 0 d 1 -d 2 ••• 0 ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• 0 0 ••• d 1 -d 2                           .

Lattices

A lattice is an additive subgroup of R n , here we recall some basic definitions and algorithms about the lattice which is usefull for the thesis.

Definition 2.2.8. consider < ., . > as an inner product.

(1) The dual

Λ * of a lattice Λ is the lattice of vectors v such that < v, w >∈ Z where w ∈ {x ∈ R n | < x, s >∈ Z f or al l s ∈ Λ}. (2) A lattice Λ is called integral if < v, w >∈ Z for all v, w ∈ Λ.
(3) A lattice Λ is called unimodular when Λ * = Λ .

(4) The volume of a lattice Λ is the volume of it's fundamental domain which is det(A) where A is the matrix of Z-basis of Λ.

It is well-known that if U 1 and U 2 are two m × n-matrices and L 1 and L 2 are the corresponding lattice generated by columns of the matrices U i , we can apply the following algorithm to find the generators of

L 1 ∩ L 2 :
Algorithm.

• Given basis

U 1 = (u 1 ,...,u n ) and V 2 = (v 1 ,..., v n ). • Take dual of U i by D(U i ) = U i U t i U i -1 = U t i -1 .
• Set K the matrix of adjunction of D(U 1 ) and D(U 2 ).

• Compute Hermite normal form of K , say H .

• Compute dual of H .

End of algorithm.

C H A P T E R

NON STANDARD HILBERT FUNCTION

Grading over abelian group

Let S = k[x 1 ,..., x n ] be a polynomial ring over field k. We first make clear our definition of grading. (1) There exists µ ∈ Q such that the vector space S µ is finite-dimensional.

(2) The only polynomials of degree zero are the constants ; i.e., S 0 = k.

(3) For all µ ∈ G , the k-vector space S µ is finite-dimensional.

(4) For all finitely generated graded modules M and degree µ ∈ G , the k-vector space M µ is finite-dimensional.

(5) The only nonnegative vector in the lattice L is 0 ; i.e., L ∩ N n = {0} . [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] The semigroup Q has no units, and no variable x i has degree zero .

Vector Partition functions

We first recall the definition of quasi-polynomials. Let d ≥ 1 and Λ be a lattice in

Z d . Definition 3.2.1. [3] A function f : Z d → Q is a quasi-polynomial with respect to Λ if there exists a list of polynomials Q i ∈ Q[T 1 ,...,T d ] for i ∈ Z d /Λ such that f (s) = Q i (s) if s ≡ i mod Λ.
Notice that Z d /Λ has | det(Λ)| elements, and that when d = 1, Λ = qZ for some q > 0, in which case f is also called a quasi-polynomial of period q.

Now assume that a positive grading of S by

Z d with Q := deg(N n ) ⊆ N d is given and that Q spans a subgroup of rank d in Z d .
In other words, the matrix A = (a i , j ) representing deg :

Z n → Z d is a d × n-
matrix of rank d with entries in N. Let a j := (a 1, j ,..., a d , j ) and

ϕ A : N d -→ N u -→ # λ ∈ N n |A.λ = u .
Equivalently, ϕ A (u) is the coefficient of t u in the formal power series

n i =1 1 (1-t a i ) .
Notice that ϕ A vanishes outside of Pos(A

) := { λ i a i ∈ R n |λ i ≥ 0, 1 ≤ i ≤ n}.
Blakley showed in [START_REF] Blakley | Combinatorial remarks on partitions of a multipartite number[END_REF] that N d can be decomposed into a finite number of parts, called chambers, in such a way that ϕ A is a quasi-polynomial of degree nd in each chamber. Later, Sturmfels in [START_REF] Sturmfels | On vectoe partition functions[END_REF] investigated these decompositions and the differences of polynomials from one piece to another.

Here we briefly introduce the basic facts and necessary terminology of vector partition functions, specially the chambers and the polynomials (quasi-polynomials) obtained from vector partition functions corresponding to a matrix A. For more details about the vector partition function, we refer the reader to [START_REF] Blakley | Combinatorial remarks on partitions of a multipartite number[END_REF][START_REF] Brion | Residue formulae, vector partition functions and lattice points in rational polytopes[END_REF][START_REF] Sturmfels | On vectoe partition functions[END_REF].

Definition 3.2.2. [53]

A polyhedral complex ℑ is a finte collection of polyhedra in R d such that 1. the empty polyhedron is in ℑ, 2. if P ∈ ℑ, then all the faces of P are also in ℑ,

3. the intersection P ∩ Q of two polyhedra P,Q ∈ ℑ is a face of both of P and of Q.

Definition 3.2.3. [22] A vector configuration in R m is a finite set A = (p j : j ∈ J ) of labeled vectors p j ∈ R m .
Its rank in the same as its rank as a set of vectors. A subconfiguration is any (labeled) subset of it .

For any subset C of the label set J we will associate the followings : 2. A subdivision ℜ refines another one ℜ (written ℜ ℜ ) if every face of ℜ is a subset of some face of

Cone A := { j ∈C λ j p j |λ j 0, ∀ j ∈ C }

ℜ

In the following we recall the definition of the chamber complex of given set A = {a 1 ,..., a n } of nonzero vectors in R d follows from [START_REF] Billera | Duality and minors of secondary polyhedra[END_REF]. Definition 3.2.6. The chamber complex Γ(A) of A is defined to be the coarsest polyhedral complex that covers Pos(A) and that refines all triangulations of A.

Note that the chamber complex is a polyhedral subdivision of a vector configuration contaning A strictly possible. Now for a given point x 0 ∈ Pos(A) we can associate the unique cell Γ(A, x 0 ) of Γ(A) which is containing x 0 . This can be written:

Γ(A, x 0 ) = {relint A |A ⊆ A, x 0 ∈ relint A } If σ ⊆ {1,.
..,n} is such that the a i 's for i ∈ σ are linearly independent, we will say that σ is independent. We set A σ := (a i )

Ω σ : G σ \{0} → Q indexed by non-trivial σ ∈ ∆(C ) such that, if u ∈ NA ∩ C , ϕ A (u) = P (u) + {Ω σ ([u] σ ).Q σ (u) : σ ∈ ∆(C ), [u] σ = 0} where [u] σ denotes the image of u in G σ . Furthermore, deg(Q σ ) = #σ -d.
Corollary 3.2.8. For each chamber C of maximal dimension in the chamber complex of A, there exists a

collection of polynomials Q τ for τ ∈ Z d /Λ such that ϕ A (u) = Q τ (u), if u ∈ NA ∩C and u ∈ τ + Λ C .
where

Λ C = ∩ σ∈∆(C ) Λ σ Proof. The class τ of u modulo Λ determines [u] σ in G σ = Z d /Λ σ .
The term of the right-hand side of the equations in the above theorem is a polynomial determined by [u] σ , hence by τ.

Notice that setting Λ for the intersection of the lattices Λ σ with σ maximal, the class of u mod Λ determines the class of u mod Λ C , hence the corollary holds with Λ in place of Λ C .

It is important to know about the relation between the partion function associated to a list of vectors in Z s and partion function associated to it's sublist, because it help us to find some recursive formula to compute the partion functions. Let X be a list of vectors a 1 , ••• , a n in Z s -{0}, in general for any aublis V of X one has the following

ϕ X (u) = ϕ X -{V } (u) * ϕ V (u)
where * denotes discrete convolution, more precisely (g * f

)(u) = µ∈Z s g (u -µ) f (µ).
Lemma 3.2.9. Let X be a list of vectors x 1 , ••• , x n in Z s -{0} then the following recursive formula hols for the vector partition function ϕ X (u)

ϕ X (u) = ∞ j =0 ϕ X -{x i } (u -j x i )
Proof. Let V = x i be a sublist of X then by using the above formula

ϕ X (u) = ϕ X -{x i } (u) * ϕ {x i } (u) = µ∈Z s ϕ X -{x i } (u -µ)ϕ {x i } (µ) = ∞ j =0 ϕ X -{x i } (u -j x i )
By the above lemma we can do the new proof for the Hilbert function of standard graded polynomial rings.

Proposition 3.2.10.

Let S = k [x 1 , ••• , x n
] be a polynomial ring over a field k and deg

x i = 1 for 1 i n. Then HF (S; m) =    m + n -1 n -1   
Proof. First we can translate the Hilbert function of S into partition function associated to X n where it consists of repetitions n-times of 1, then by using induction on n. For the case of n=1 is clear. Now by lemma3.2.9 we have

HF (S; m) = ϕ X n (m) = ∞ j =0 ϕ X n-1 (m -j )
we now use our assumption that the formula is true for n -1 and Pascal formula

ϕ X n (m) = ∞ j =0    m -j + n -2 n -2    = ∞ j =0    m -j + n -1 n -1    -    m -j + n -2 n -1    =    m + n -1 n -1   

Lattice points problem and Barvinok algorithm

Definition 3.3.1. [START_REF] Barvinok | An algorithmic theory of lattice points in polyhedra[END_REF] A rational polyhedron P ⊂ R d is the set of solutions of a finite system of linear inequalities with integer coefficient :

P = {x ∈ R d :< c i , x > β i f or i = 1, ••• , m} A bounded
#(nP ∩ Z d ) = q(n) wher e q(n) = vol (P )n d + a d -1 n d -1 + ••• + a 1 x + 1.
In the general case for any rational polyhedron P ⊂ R n we consider following generationg function:

f (P, x) = m∈P ∩Z n x m where m = (m 1 , ••• , m n ) and x m = x m 1 1 ••• x m n n .

Valuations and generating functions of rational polyhedra

To explain berifly the Barvinok method first we should define the vertex cone and generating function associated to each polytope. Theorem 3.3.7. [START_REF] Barvinok | An algorithmic theory of lattice points in polyhedra[END_REF] There is a map Φ which , to each rational polyhedron P ⊂ R d associates a rational function f (P ; x) in the d complex variables x ∈ C d such that the following properties are stisfied:

(1) The map Φ is a valuation.

(2) If u + P is a translation of P by an integer vector u ∈ Z d , then f (u + P ; x) = x u f (P ; x).

(3) We have

f (P, x) = m∈P ∩Z n x m
for any x ∈ C d such that the series converges absolutely.

(4) If P contains a straight line then f (P, x) ≡ 0.

Definition 3.3.8. [START_REF] Barvinok | An algorithmic theory of lattice points in polyhedra[END_REF] Let P ⊂ R d be a polyhedron and let v ∈ P be a vertex of P. The tangent cone K = cone(P, v) of P at v is defiend as follows:

suppose that

P = {x ∈ R d :< c i , x > β i f or i = 1, ••• , m} is a representation of P , where c i ∈ R d and β i ∈ R. Let I v = {i :< c i , v >= β i } be the set of constraints that are active on v. Then K = cone(P, v) = {x ∈ R d :< c i , x > β i f or i ∈ I v } Lemma 3.3.9. [4]Let K ⊂ R d be a pointed rational cone. Then K = co(w 1 , ••• , w n ) for some w 1 , ••• , w n ∈ Z d -{0}. Let us define W K = {x ∈ C d | |x w i | < 1 f or i = 1, ••• , n}.
Then W K is a non-empty open set and for every x ∈ W K , the series m∈K ∩Z n x m converges to a rational function f (K , x) of the type

f (K , x) = i ∈I i x v i (1 -x u i 1 ) •••(1 -x u id )
where i ∈ {1, -1}, v i ∈ Z d and u i j ∈ Z d -{0} for all i and j . Now by Brion's theorem, the generating function of the polytope P is equal to the sum of the generating functions of its vertex cones, more precisely

f (P ; x) = m∈P ∩Z n x m = v∈Ω(P ) f (K ; v)
where Ω(P ) is the set of vertices of P .

Example 3.3.10. [START_REF] De Loera | Effective lattice point counting in rational convex polytopes[END_REF]Consider the quadrilateral with the vertex V 1 = (0, 0), V 2 = (5, 0), V 3 = (4, 2), and V 4 = (0, 2). Then wen obtain the following rational functions for each vertex :

f (K ,V 1 ) = 1 (1 -x 1 )(1 -x 2 ) , f (K ,V 1 ) = (x 5 1 + x 4 1 x 2 ) (1 -x -1 1 )(1 -x 2 2 x -1 1 ) f (K ,V 3 ) = (x 4 1 x 2 + x 4 1 x 2 2 ) (1 -x -1 1 )(1 -x 1 x -2 2 ) , f (K ,V 4 ) = x 2 2 (1 -x -1 2 )(1 -x 1 )

Decomposing a rational cone into unimodular cones

In order to finding the generation function of arbitrary pointed cones Stanly give the formula by using a triangulation of a rational cone into simplicial cones but instead in 1994 Barvinok proved the general fact that every rational poolyhedral cone can be triangulated into uimodular cones as follows:

Theorem 3.3.11. [START_REF] Barvinok | An algorithmic theory of lattice points in polyhedra[END_REF] Fix the dimension d . Then , there exists a polynomial time algorithm, for a rational

polyhedral cone K ⊂ R d , which computes unimodular cones K i , i ∈ I = {1, ••• , r } , and numbers ∈ {-1, 1}
such that

[K ] = i ∈I [K i ].
Remark 3.3.12. By having the above decomposition of cones we can write f (K , x) = i ∈I i f (K i , x), as we have an explicit formula for the unimodular case so we can calculate an explicit formula for rational cones , it is a key idea of the section3.5.

Theorem 3.3.13. [7] Fix the dimension d . There exists a polynomial time algorithm, for a rational poly-

hedron P ⊂ R d , P = {x ∈ R d :< c i , x > β i f or i = 1, ••• , m} wher e c i ∈ Z d and β i ∈ Q computes the generation function f (P, x) = m∈P ∩Z n x m as follows f (P, x) = i ∈I i x a i (1 -x b i 1 ) •••(1 -x b id ) b 1 b 2 u 1 1 v 1 L 3 2 E 2 E 1

Decomposition of two-dimensional cones and continued fraction

Here we recall the algorithm of continued fraction expansion of a real number r as follows:

we can write r as r = r + ρ(r ) where 0 ρ(r ) < 1 and let r 0 = r , if ρ(r ) = 0 we stop the algorithm , if not we put a as 1 ρ(r ) then we repeat as above. At the end we represent by it's continued fraction as Theorem 3.3.15. [START_REF] Barvinok | Integer points in polyhedra[END_REF]Let K ⊂ R 2 be a cone generated by vectors (1, 0) and (q, p) where q and p are coprime positive integers. Let

r = [r 0 ; r 1 , ••• , r n , •••]
p p = [a 0 , ••• , a n ] , we define the cone K i for i = -1, 0, ••• , n as follows : consider p i q i = [a 0 , ••• , a i ] f or i = 0, ••• , n.
Let K -1 be the cone generated by (1, 0) and (0, 1) , K 0 as the cone generated by (0, 1) and (1, p 0 ) , and K i as the cone generated by (q i -1 , p i -1 ) and (q i , p i ) for i = 1, ••• , n . Then K i are unimodular cones for i = -1, 0, 1, ••• , n and we can write K as follows :

[K ] =              n i =-1 (-1) i +1 [K i ] if n i s od d n i =-2 (-1) i +1 [K i ] if n i s even
where K -2 is the ray emanating from the origin in the direction of (q n , p n ).

Remark 3.3.16. It is not hard to see that an arbitrary 2-dimensional cones can be represented as a combination of two 2-dimensional cones each of them generated by (1, 0) and some other integer vector, and one 1-dimensional cone.

(1) In the above theorem if q < 0, we let K 1 be the cone generated by (1, 0) and (-q, -p) then the indicators (2) I the above theorem if p < 0 we let K 1 be the cone generated by (1, 0) and (q, -p)

then f (K 1 ,(x 1 , x 2 )) = f (K 2 ,(x 1 , x -1 2 )) .

Hilbert functions of non-standard bigraded rings

Let S = k[y 1 ,..., y m ] be a Z d -1 -graded polynomial ring over a field and let I = ( f 1 ,..., f n ) be a graded ideal, with f i homogeneous of degree d i . To get information about the behavior of i -syzygy module of I t as t varies, we pass to Rees algebra

R I = ⊕ t ≥0 I t which is a (Z d -1 × Z)-graded algebra such that (R I ) (µ,n) = (I n ) µ .
Recall that R I is a graded quotient of R := S[x 1 ,..., x n ] with grading extended from the one of S by setting deg(a) := (deg(a), 0) for a ∈ S and deg(x j ) := (d j , 1) for all j . As noticed in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF], if

G • is a Z d -graded free R-resolution of R I , then, setting B := k[x 1 ,..., x n ] = R/(y 1 ,..., y m ), Tor S i (I t , k) µ = H i (G • ⊗ R B ) (µ,t ) .
The complex G • ⊗ R B is a Z d -graded complex of free S-modules. Its homology modules are therefore finitely generated Z d -graded S-modules, on which we will apply results derived from the ones on vector partition functions describing the Hilbert series of S.

As an example, we describe the chamber complex associated to the matrix corresponding to the degrees (d i , 1), when d = 1 (i.e. d i ∈ N). Assume that the number of distinct d i 's is r ≥ 2. Then there exist a finite index sublattice L of Z 2 and collections of polynomials Q i j of degree n -2 for 1 ≤ i ≤ r -1 and 1 ≤ j ≤ s such that for any (µ, ν) ∈ Z 2 ∩R i and µ, ν ≡ g j mod L in Z 2 /L := {g 1 ,..., g s },

HF (B,(µ, ν)) = Q i j (µ, ν)
where R i is the convex polyhedral cone generated by linearly independent vectors {(d i , 1), (d i +1 , 1)}.

Furthermore, Q i j (µ, ν) = Q i j (µ , ν ) if µ -νd i ≡ µ -ν d i mod (det(L)). Proof. Let A =    d 1 ... d n 1 ... 1   
be a 2 × n-matrix corresponding to degrees of T i .

The Hilbert function in degree u = (µ, ν) is the number of monomials T

α 1 1 ...T α n n such that α 1 (d 1 , 1) + ... + α n (d n , 1) = (µ, ν).
This equation is equivalent to the system of linear equations

A.        α 1 . . . α n        = ( µ ν ).
In this sense HF (B,(µ, ν)) will be the value of vector partition function at (µ, ν). Assume that (µ, ν) belongs to the chamber R i which is the convex polyhedral cone generated by {(d i ,1),(d i +1 , 1)}. By 3.2.8, we know that for (µ, ν) ∈ R i and (µ, ν) ≡ g j mod (det L),

ϕ A (µ, ν) = Q i j (µ, ν). (3.4.1)
Notice that in the proposition 3.4.2, if moreover we suppose that d i = d j for all i = j , then the Hilbert function in degree (µ, ν) will also be the number of possible ways to reach from (0, 0) to (µ, ν) in Z 2 but it is not necessarily correct when we have equalities between some of degrees. For example if one has

d i = d i +1 < d i +2
, so the independent sets of vectors {(d i ,1),(d i +2 , 1)} and {(d i +1 , 1), (d i +2 , 1)} generate the same chamber and the number of possible ways to reach from (0, 0) to (µ, ν) is less than HF(B,(µ, ν)).

In the following example, we are going to give a formula for Hilbert function of a non-standard graded polynomial ring in the case of three indeterminates which is a special case of formula done by Xu in [START_REF] Xu | An explicit formulation for two dimensional vector partition functions. Integer points in polyhedra-geometry, number theory, representation theory, algebra, optimization[END_REF].

T i = (d i ,1) for 1 ≤ i ≤ 3 such that d i -d i +1 0 for i = 1, 2. Set Y i j = d i -
Ω i = {(µ, ν) | νd i > µ > νd i +1 } for i = 1, 2.
Then for (n 1 , n 2 ) belonging to the positive cone generated by vectors {(d 1 , 1), (d 2 ,1),(d 3 , 1)}. When (n 1 , n 2 ) t ∈

Ω 1 ∩ Z 2 , it is proved in [54, Theorem 4.3] that HF (B,(n 1 , n 2 )) = n 2 d 1 -n 1 Y 12 Y 13 + 1 -(f12Y13+g12Y23) -1 (n2(f12d1+g12d2)-n1(f12+g12) Y 12 -(f13Y12+g13Y23) -1 (n2(f13d1+g13d3)-n1(f13+g13)) Y 13 .

Explicit formula for non-standard Hilbert function

In this section we want to give an explicit formula for non standard Hilbert functions of polynimial ring in low dimensions by the using theory of lattice points in the convex polytope.

Variable Polytopes of partition function

Let e i be the standard basis of the space R r for 1 i r and linear map f : R r → R 2 defined by f

(e i ) = v i .
Let a ∈ R 2 , we define the following convex polytope:

P (a) := f -1 (a) ∩ R r 0 = {x ∈ R r |Ax = a; x 0}
Where A is the matrix of f . Proposition 3.5.1. [START_REF] De Concini | Topics in hyperplane arrangements, polytopes and boxsplines[END_REF] Now we consider the polytope

Let A = {a 1 , ••• , a n } be a set of vectors in R d . If b is in the interior of Pos(A) := { n i =1 λ i a i ∈ R d |λ i ≥ 0, 1 ≤ i ≤ n}, the polytope P (b) has dimension n -d . Definition 3.5.2. Let A = {a 1 , ••• , a n } be a set of vectors in R d . 1. Let b ∈ Pos(A). A basis a i 1 , ••• , a i s
P (b) such that A is    d 1 d 2 d 3 d 4 1 1 1 1    for d 1 < d 2 < d 3 < d 4 then by
the method of Barvinok we want to calculate the generating function of the tangent cones of the polytopes depending on b, before that we should mention that the polytope P (b) associated to the matrix A is not full dimensional so to use the Barvinok metod we need to transform P (b) to polytope Q which is full dimensional and the integer points of Q are in one-to-one correspondence to the integer points of P (b). The following procedure describes how it can be done:

1. let P = {x ∈ R n |Ax = a, B
x b} be a polytope related to full row-rank d × n matrix A.

Find the generators {g

1 , ••• , g n-d } of the integer null-space of A.
3. Find integer solution x 0 to Ax = a.

Substituting the general integer solution

x = x 0 + n-d i =1 β i g i into the inequalities B x b.
5. By Substitution of (4) we arrive at a new system C β c which defines the new polytope

Q = {β ∈ R n-d |C β c}.
By using above procedure we will associat to the polytope P (b) the full dimensional polytope Q in the following lemma:

Lemma 3.5.4. Let A =    d 1 d 2 d 3 d 4 1 1 1 1    for d 1 < d 2 < d 3 < d 4
then there is a one to one correspondence to the integer points of P (b) and Q ⊂ R 2 and we have the followings about Q:

1. Q = (λ 1 , λ 2 ) ∈ R 2 |λ 1 (d 2 -d 1 ) 0; λ 2 (d 2 -d 1 ) 0; λ 1 (d 1 -d 4 ) + λ 2 (d 1 -d 3 ) + d 1 b 2 -b 1 d 2 -d 1 0; λ 1 (d 4 -d 2 ) + λ 2 (d 3 -d 2 ) + b 1 -d 2 b 2 d 2 -d 1 0 f or b 1 , b 2 > 0 ,
2. with the following vertices :

q 1 = ( d 3 b 2 -b 1 (d 2 -d 1 )(d 4 -d 3 ) ,( b 1 -d 4 b 2 (d 2 -d 1 )(d 4 -d 3 ) ) , q 2 = ( d 2 b 2 -b 1 (d 2 -d 1 )(d 4 -d 2 ) , 0) , q 3 = (0, d 2 b 2 -b 1 (d 2 -d 1 )(d 3 -d 2 ) ) , q 4 = ( b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 4 ) , 0) , q 5 = (0, b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 3 )
) , q 6 = (0, 0),

The generation function of Q in the first chamber C

1 : f C 1 (Q, x) = 1 (1-x -1 1 )(1-x -1 2 ) + x s 1 1 (1-x 1 )(1-x -1 2 ) - x s 1 1 x -( a 0 s 1 +a 0 s 1 ) 2 (1-x -1 2 )(1-x 1 x -a 0 2 ) + x ( (a 0 a 1 +1)s 1 -a 1 a 0 s 1 ) 1 x -(a 0 (a 0 a 1 +1)s 1 -(a 0 a 1 +1) s 1 ) 2 (1-x 1 x -a 0 2 )(1-x a 1 1 x -(a 1 a 0 +1) 2 ) + x -s 2 2 (1-x -1 2 )(1-x 1 x -a 0 2 ) - x ( -a 1 s 2 +a 1 s 2 ) 1
x

-(a 0 -a 1 s 2 +(a 0 a 1 +1) s 2 ) 2 (1-x 1 x -a 0 2 )(1-x a 1 1 x -(a 1 a 0 +1) 2 )
Where s

1 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 4 ) and s 2 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 3 ) . Proof. (1) It is not hard to see that a generators of integer null-space of A =    d 1 d 2 d 3 d 4 1 1 1 1    are -→ g 1 = (d 2 -d 4 , d 4 -d 1 , 0, d 1 -d 2 ) and -→ g 2 = (d 2 -d 3 , d 3 -d 1 , d 1 -d 2 ,0) and ( b 1 -d 2 b 2 (d 1 -d 2 ) , d 1 b 2 -b 1 (d 1 -d 2 )
, 0, 0) is the solution of linear system then considering the five steps in which describe before this lemma we can get supporting half planes of Q. (2) can easily calculated from (1). As we suppose that b ∈ C 1 then only vertices Q 6 ,Q 4

and Q 5 are active, then we associate to each one a tangent cone as follows:

Cone(Q, q 5 ) = co((1, 0), (d 3 -d 1 , d 1 -d 4 )), Cone(Q, q 4 ) = co((0, 1), (d 1 -d 3 , d 4 -d 1 )), Cone(Q, q 4 ) = co((-1, 0), (0, -1)).
The first two tangent cones are necessarily unimodular so as our cone are 2-dimensional we can decompose it to unimodular cone by continued fraction. For simplicity of calculations suppose that

d 4 -d 1 d 3 -d 1 = [a 0 ; a 1 ] = a 0 + 1 a 1 then [Cone(Q, q 5 )] = [K -1 = co((1, 0), (0, 1))] + [K 0 = co((0, 1), (1, a 0 ))] + [K 1 = co((1, a 0 ), (a 1 , a 0 a 1 + 1))]
d i = d j for 1 i 4 . Suppose that d 4 -d 1 d 3 -d 1 = a 0 + 1 a 1 and set s 1 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 4 ) and s 2 := b 1 -d 1 b 2 (d 2 -d 1 )(d 1 -d 3 )
. Then the Hilber function of B at degree (b 1 , b 2 ) ∈ C 1 given by following formula:

HF (B ;(b 1 , b 2 )) = (a 0 +2)(a 0 +1) 2(a 0 -1) + 4a 0 (a 0 +2)-3a 2 0 12 + a 0 (a 0 +2) 2(a 0 -1) + (a 0 s 1 +2)(a 0 s 1 +1) 2(a 0 -1)
+ a 0 (a 0 s 1 +2) 2(a 0 -1) + 4a 0 (a 0 +2)-3a 2 0 12 4a 0 (a 0 +2)-3a 2 0 6 + a 0 (a 0 s 1 +2 a 0 s 1 +1)(a 0 s 1 +2 a 0 s 1 +1)

2(a 0 -1)

+ a 0 (a 0 +2)(a 0 +1) 2(a 0 -1)

+ a 0 (a 0 +2) 2(a 0 -1) +(a 0 +2)(a 0 s 1 +2 a 0 s 1 )+ a 0 (a 0 s 1 +2 a 0 s 1 )

2(a 0 -1) + (4a 0 (a 0 +2)-3a 2 0 )(2a 0 a 1 +3) 12(2a 0 a 1 +1) + (a 0 (a 0 a 1 +1)s 1 )(a 0 (a 0 a 1 +1)s 1 +1) 2(a 0 -1) + (4(a 0 a 1 +2)(a 0 a 1 +4)-3(a 0 a 1 +2) 2 )((2a 0 -1))

12(a 0 -1)

+ (a 0 a 1 s 1 +2 s 1 +a 0 a 1 +2)(a 0 a 1 s 1 +2 s 1 +a 0 a 1 +1)(a 2 0 a 1 +2a 0 -1) (a 0 -1)(a 0 a 1 +1) + (a 0 a 1 s 1 +2 s 1 +a 0 a 1 +2)a 0 2(a 0 -1) + (a 0 a 1 s 1 +2 s 1 +a 0 a 1 +2)((a 0 (a 0 a 1 +1)s 1 )) 1 + (a 0 a 1 s 1 +2 s 1 +a 0 a 1 +2)(a 0 a 1 +2) 2(a 0 a 1 +2) + (a 2 0 (a 0 a 1 +1)s 1 ) 2(a 0 -1)
a 0 (a 0 a 1 +2))

2(a 0 -1)(a 0 a 1 +1) + (a 0 (a 0 a 1 +1)s 1 )(a 0 a 1 +2)

2(a 0 a 1 +1) + a 0 (a 0 +2)(a 0 +1) 2(a 0 -1)

+ a 0 (2 s 2 )(2 s 2 +1)
2(a 0 -1) + 4a 0 (a 0 +2)-3a 2 0 a 0 (a 0 +2)

2(a 0 -1) + 2(a 0 +2) s 2 1 + 2a 0 s 2 2(a 0 -1) + (a 0 a 1 +1)(a 0 a 1 +2)(a 2 0 a 1 +2a 0 -1) 2(a 0 -1)(a 0 a 1 +1) + (4a 0 (a 0 +2)-3a 2 0 )(2a 0 a 1 +3) 12(2a 0 a 1 +1) + (2a 0 -1)(4(a 0 a 1 +2)(a 0 a 1 +4)-3(a 0 a 1 +2) 2 )((2a 0 -1))
12(a 0 -1)

+ (a 0 a 1 s 2 +2 s 2 +a 0 a 1 s 2 )(a 0 a 1 s 2 +2 s 2 +a 0 a 1 s 2 +1)(a 2 0 a 1 +2a 0 -1) 2(a 0 -1)(a 0 a 1 +1) a 0 (a 0 a 1 +2) 2(a 0 -1) + (a 0 a 1 +2) 2
2(a 0 a 1 +1) + (a 0 a 1 +2)(a 0 a 1 s 2 +2 s 2 +a 0 a 1 s 2 ) 1 + a 0 (a 0 a 1 +2)) 2(a 0 -1)(a 0 a 1 +1) + a 0 (a 0 a 1 s 2 + s 2 +a 0 a 1 s 2 )

2(a 0 -1

(a 0 a 1 +2)(a 0 a 1 s 2 +2 s 2 +a 0 a 1 s 2 ) 2(a 0 a 1 +2)
.

Proof Choose the vector η = (a 0 , 2) which is not orthogonal to any generators of the vertex cones of Q and variable substituition x i = (s + 1) η i for i = 1, 2 then we obtain:

f C 1 (Q, ((s + 1) a 0 ), (s + 1) 2 )) = (s+1) a 0 +2 (1-(s+1) a 0 )(1-(s+1) 2 ) - (s+1) a 0 s 1 +2 (1-(s+1) a 0 )(1-(s+1) 2 ) + (s+1) a 0 +2-2 a 0 s 1 -a 0 s 1 (1-(s+1) a 0 )(1-(s+1) 2 ) + (s+1) -a 0 (a 0 a 1 +1)s 1 +a 0 a 1 s 1 +2 s 1 +a 0 a 1 +a 0 +2 (1-(s+1) a 0 )(1-(s+1) a 0 a 1 +2 ) -(s+1) a 0 +2 +2 s 2 (1-(s+1) a 0 )(1-(s+1) 2 ) + (s+1) -a 0 a 1 s 2 -a 0 a 1 s 2 -2 s 2 +a 0 a 1 +a 0 +2 (1-(s+1) a 0 )(1-(s+1) a 0 a 1 +2 )
Now we using following general expansions:

(s + 1) n ≡ 1 + ns + n(n -1) 2 s 2 mod s 3 and 1 s + 2 ≡ 1 2 - 1 4 s + 1 4 s 2 mod s 3
we obtain:

(s + 1) a 0 +2 (1 -(s + 1) a 0 )(1 -(s + 1) 2 ) ≡ (1 + (a 0 + 2)s + (a 0 + 2)(a 0 + 1)s 2 2 )( 1
a 0 -1 + a 0 s 2(a 0 -1) + (4a 0 (a 0 + 2) -3a 2 0 )s 2 12 ) , (s + 1) a 0 s 1 +2 (1 -(s + 1) a 0 )(1 -(s + 1) 2 ) ≡ (1+(a 0 s 1 +2)s+( (a 0 s 1 + 2)(a 0 s 1 + 1) 2 )s 2 )( 1
a 0 -1 + a 0 s 2(a 0 -1)
+ (4a 0 (a 0 + 2) -3a 2 0 )s 2 12)

,

(s + 1) a 0 +2-2 a 0 s 1 -a 0 s 1 (1 -(s + 1) a 0 )(1 -(s + 1) 2 ) ≡ (1 + (a 0 + 2)s + (a 0 + 2)(a 0 + 1)s 2 2 )( 1
a 0 -1 + a 0 s 2(a 0 -1) + (4a 0 (a 0 + 2) -3a 2 0 )s 2 12 ) (1+(a 0 s 1 +2 a 0 s 1 )s+ (1 + a 0 s 1 + 2 a 0 s 1 )(a 0 s 1 + 2 a 0 s 1 ) 2 s 2 )( 1
a 0 -1 + a 0 s 2(a 0 -1) + (4a 0 (a 0 + 2) -3a 2 0 )s 2 12 )
, (s + 1) -a 0 (a 0 a 1 +1)s 1 +a 0 a 1 s 1 +2 s 1 +a 0 a 1 +a 0 +2

(1

-(s + 1) a 0 )(1 -(s + 1) a 0 a 1 +2 ) ≡ (1 + (a 0 a 1 s 1 + 2 s 1 + a 0 a 1 + a 0 + 2)s+ (a 0 a 1 s 1 + 2 s 1 + a 0 a 1 + a 0 + 2)(a 0 a 1 s 1 + 2 s 1 + a 0 a 1 + a 0 + 1) 2 s 2 )
( 1

a 0 -1 + a 0 s 2(a 0 -1) + (4a 0 (a 0 + 2) -3a 2 0 )s 2 12
)(1+(a 0 (a 0 a 1 +1)s 1 )s+ (a 0 (a 0 a 1 + 1)s 1 )(a 0 (a 0 a 1 + 1)

s 1 + 1) 2 s 2 ) ( 1 (a 0 a 1 + 1) + (a 0 a 1 + 2) 2(a 0 a 1 + 1) s + 4(a 0 a 1 + 2)(a 0 a 1 + 4) -3(a 0 a 1 + 1) 2 12 s 2 ) , (s + 1) a 0 +2 + 2 s 2 (1 -(s + 1) a 0 )(1 -(s + 1) 2 ) ≡ (1 + (a 0 + 2)s + (a 0 + 2)(a 0 + 1) 2 s 2 )( 1
a 0 -1 + a 0 2(a 0 -1) s + 4a 0 (a 0 + 2) -3a 2 0 12 s 2 ) (1 + 2 s 2 s + (2 s 2 )(2 s 2 + 1) 2 s 2 )
and

(s + 1) -a 0 a 1 s 2 -a 0 a 1 s 2 -2 s 2 +a 0 a 1 +a 0 +2 (1 -(s + 1) a 0 )(1 -(s + 1) a 0 a 1 +2 ) ≡ (1+(a 0 a 1 +2)s+ (a 0 a 1 + 2)(a 0 a 1 + 1) 2 s 2 )( 1
a 0 -1 + a 0 2(a 0 -1) + 4a 0 (a 0 + 2) -3a 2 0 12 s 2 ) ( 1
a 0 a 1 + 1 + a 0 a 1 + 2 2(a 0 a 1 + 2) s + 4(a 0 a 1 + 2)(a 0 a 1 + 4) -3(a 0 a 1 + 2) 2 12 s 2 ) (1 + (a 0 a 1 s 2 + a 0 a 1 s 2 + 2 s 2 )s + (a 0 a 1 s 2 + a 0 a 1 s 2 + 2 s 2 )(a 0 a 1 s 2 + a 0 a 1 s 2 + 2 s 2 + 1) 2 s 2 )
So the number of lattice points in the polytope Q is given by the sum of the coefficients of s 2 which is the final formula.

In the case where M = n 0 I n the polynomials

P i (k) = β i ( R I k ) = dim Tor R i ( R m , R I k ) are called the
Kodiyalam Polynomials of I for k 0 and i 0 .

One of the above theorem's outcomes is about projective and injective dimensions of R I k which is proved first by Brodmann [START_REF] Brodmann | The asymptotic nature of the analytic spread[END_REF] .

P 1 (I )(k) = (k + 1) 2 , P 2 (I )(k) = ( 5 2 k + 7 2 )k , P 3 (I )(k) = 3 2 k(k + 1) .
Kodiyalam asked in [START_REF] Kodiyalam | Asymptotic behaviour of Castelnuovo-Mumford regularity[END_REF] that " is it true that polynomials P i (I )(n) for 0 either vanishes or has degree l (I ) -1 ? " , then J. Herzog and V. Welker proved in [START_REF] Herzog | The Betti polynomials of powers of an ideal[END_REF] the following result about degrees of P i (I )

Proposition 4.1.4. l -1 = deg P 1 (I )(k) P 2 (I )(k) ••• P n (I )(k).
In fact, the Kodiyalam polynomials is explained asymptotic behavior of total Betti numbers of powers of ideals. More generally, in the next section, we will study the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals.

The general case

Before studying the graded Betti numbers of powers of ideals let me recall the result in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] about the importent case where the generators of the ideal I have the same degree.

(M I t 1 1 ••• I t s s , k) µ = (B i ) µ,t 1 e 1 +•••+t s e s .
The following result applied to B i will then give the asymptotic behavior of Betti numbers. In the particular case of one Z-graded ideal, we will use it to give a simple description of this eventual behaviour.

Let ϕ :

Z n → Z d with ϕ(N n ) ⊆ N d be a positive Z d -grading of R := k[T i , j ]. Set Z n := n i =1 Ze i ,
Λ := ∩ e∈E Λ e , s Λ : Z d can / / Z d /Λ .
Denote by C i , i ∈ F , the maximal cells in the chamber complex associated to ϕ. One has

C i = {ξ | H i , j (ξ) 0, 1 j d } where H i , j is a linear form in ξ ∈ Z d .
∆ u = {x | H i , j (x) a u,i , j , ∀(i , j ) ∈ G u } ⊆ R d for u ∈ U , U finite, with a u,i , j = H i , j (a) for a ∈ ∪ Supp Z d (Tor R (B, k)), G u ⊂ F × {1,...,d } and polynomials P u,τ for u ∈ U and τ ∈ Z d /Λ such that: dim k (B ξ ) = P u,s Λ (ξ) (ξ), ∀ξ ∈ ∆ u , and dim k (B ξ ) = 0 if ξ ∈ ∪ u∈U ∆ u .
Proof. By Proposition 2.1.

i ,τ such that dim k R ξ-a = Q i ,τ (ξ -a) if ξ -a ≡ τ mod (Λ).
Hence, setting Q i ,a,τ (ξ) := Q i ,τ+a (ξa), one gets the conclusion with

P u,τ = (i ,a)∈u c a Q i ,a,τ .
Remark 4.2.3. The above proof shows that if one has a finite collection of modules B i , setting A := ∪ i , Supp Z d Tor R (B i , k), there exist convex polyhedral cones ∆ u as above on which any B i has its Hilbert function given by a quasi-polynomial with respect to the lattice Λ.

Let S = k[y 1 ,..., y m ] be a Z p -graded polynomial ring over a field. Assume that deg(y j ) ∈ N p for any j , and let I i = ( f i ,1 ,..., f i ,r i ) be ideals, with f i , j homogeneous of degree d i , j .

Consider

R := k[T i , j ] 1≤i ≤s,1≤ j ≤r i , set deg(T i , j ) = (deg( f i , j
), e i ), with e i the i -th canonical generator of Z s and the induced grading ϕ :

Z r 1 +•••+r s → Z d := Z p × Z s of R.

The case of one graded ideal on a positively Z-graded ring

We now explain in detail an important special case: one ideal in a positively Z-graded polynomial ring over a field. We will use the following elementary lemma.

β j 2 ) ∈ R 2 for 1 ≤ j ≤ N , consider the half-lines L j i := {(β j 1 , β j 2 ) + λ(d i , 1), λ ∈ R ≥0 } and set L j i (t ) := L j i ∩ {y = t }.
Then there exist a positive integer t 0 and permutations σ i , for i = 1, ••• , r , in the permutation group S N such that, for all t ≥ t 0 , the following properties are satisfied :

(1) L

σ i (1) i (t ) L σ i (2) i (t ) ••• L σ i (N ) i (t ) for 1 i r , (2) L σ i (N ) i (t ) L σ i (1) i +1 (t ).
Moreover t 0 can be taken as the biggest second coordinate of the intersection points of all pairs of half lines.

Proof. If two half-lines L j i and L v u intersect at a unique point A(x A , y A ), then

y A = det    β v 1 d u β v 2 1    -det    β j 1 d i β j 2 1    d i -d u .
Choose t 0 as the max of y A , A running over the intersection points. For t ∈ [t 0 , +∞[ the ordering of the intersection points L j i (t ) on the line {y = t } is independent of t . Furthermore, as the d i 's are strictly increasing (2) holds, which shows (1) as the ordering is independent of t . Notice further that Λ := ∩ i < j Λ i , j is a sublattice of Λ for any .

CHAPTER 4. ASSYMPTOTIC BEHAVIOR BETTI NUMBER OF POWERS OF IDEALS

There exist, t 0 , m, D ∈ Z, linear functions L i (t ) = a i t + b i , for i = 0,...,m, with a i among the degrees of the minimal generators of I and b i ∈ Z, and polynomials Q i , j ∈ Q[x, y] for i = 1,...,m and j ∈ 1,...,D, such that, for t ≥ t 0 , (i) L i (t ) < L j (t ) ⇔ i < j , (ii) If µ < L 0 (t ) or µ > L m (t ), then Tor S i (I t , k) µ = 0.

(iii) If L i -1 (t ) ≤ µ ≤ L i (t ) and a i tµ ≡ j mod (D), then dim k Tor S i (I t , k) µ = Q i , j (µ, t ).

Proof. We know from [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] that M := Tor S i (I t , k) is a finitely generated Z 2 -graded module over R. Then it follows from Proposition 4. In particular, it provides a B -structure on ⊕ t Tor S i (M I t , k) making it a finitely generated B -module.

Slightly more generally, it was showed in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] that the following holds. There exist, t 0 , m, D ∈ Z, linear functions L i (t ) = a i t + b i , for i = 0,...,m, with a i among the degrees of the minimal generators of I and b i ∈ Z, and polynomials Q i , j ∈ Q[x, y] for i = 1,...,m and j ∈ 1,...,D, such that, for t ≥ t 0 , (i) L i (t ) < L j (t ) ⇔ i < j , (ii) If µ < L 0 (t ) or µ > L m (t ), then Tor S i (I t , k) µ = 0.

(iii) If L i -1 (t ) ≤ µ ≤ L i (t ) and a i tµ ≡ j mod (D), then dim k Tor S i (I t , k) µ = Q i , j (µ, t ).

Proof. By the above the theorem we know that Tor R i (M R I , k) is finitely generated k[T 1 ,...,T r ]-module then the result follows from proposition 4.3.2.

structure of Tor module of Hilbert filtrations

To study blowup algebras, Northcott and Rees defined the notion of reduction of an ideal I in a commutative ring R. An ideal J ⊆ I is a reduction of I if there exists r such that J I r = I r +1 (equivalently this hold for r 0) . An impotent fact about reduction ideal J of I is that it is equivalent to ask that

R J = ⊕ n J n → R I = ⊕ n I n
is a finite morphism. Okon and Ratliff in [START_REF] Okon | Reductions of filtrations[END_REF] extended the above notion of reduction to the case of filtrations by setting the following definition: Definition 5.2.1. If R is a ring and I and J be ideals in R, then:

(1) A filtration ϕ = {ϕ(n)} n≥0 on R is a decreasing sequence of ideals ϕ(n) of R such that ϕ(0) = Rand ϕ(m)ϕ(n) ⊆ ϕ(m + n) for all nonnegative integers m and n.

(2) If ϕ and γ are filtrations on R, then ϕ = γ in case ϕ(n) = γ(n) for all n ≥ 0, and ϕ ≤ γ in case ϕ(n) ⊆ γ(n) for all n ≥ 0.

(3) If ϕ and γ are filtrations on R, then ϕ is a reduction of γ in case ϕ ≤ γ and there exists a positive integer d such that γ(n) = d i =0 ϕ(ni )γ(i ) for all n ≥ 1.

(4) Let I be an ideal of R and ϕ is filtration on R, then ϕ is called I -good filtration if I ϕ i ⊆ ϕ i +1 for all i ≥ 0 and ϕ n+1 = I ϕ n for all n 0.

(2) There exist, t 0 , m, D ∈ Z, linear functions L i (t ) = a i t + b i , for i = 0,...,m, with a i among the degrees of the minimal generators of I and b i ∈ Z, and polynomials Q i , j ∈ Q[x, y] for i = 1,...,m and j ∈ 1,...,D, such that, for t ≥ t 0 , (i) L i (t ) < L j (t ) ⇔ i < j , (ii) If µ < L 0 (t ) or µ > L m (t ), then Tor S i (ϕ(t ), k) µ = 0.

(iii) If L i -1 (t ) ≤ µ ≤ L i (t ) and a i tµ ≡ j mod (D), then dim k Tor S i (ϕ(t ), k) µ = Q i , j (µ, t ).

Proof. Let F • be a Z × Z-graded minimal free resolution of R ϕ over R. Each F i is of of finite rank due to the Noetherianity of A . The graded stanf F t • := (F • ) * ,t is a Z-graded free resolution of ϕ(t ) over S = R ( * ,0) . Thus,

Tor S i (ϕ(t ), k) = H i (F t • ⊗ S k).
Moreover, taking homology respects the graded structure, and therefore,

H i (F t • ⊗ S k) = H i (F • ⊗ R R/m + nR) ( * ,t ) ,
where n = (x 1 ,..., x n ) is the homogeneous irrelevant ideal in S. So it follws that Tor R j (R I , k) is finitely generated graded k[T 1 ,...,T r ]-module .The second fact comes from proposition 4.3.2.

This in particular applies to the ideals when ever (A, m, k) is local Noetherian ring and S be a graded local Noetherian algebra over A:

• If I be a graded ideal of S and S be analytically unramified ring without nilpotent elements then ϕ(n) = I n is I -good filtration then result follows from theorem 5.2.3.

• If I be a graded ideal of S then by theorem 2.1.20 the filtration ϕ(n) = I n is I -good filtration then result follows from theorem 5.2.3.

.

Theorem 1 . 0 . 2 .

 102 (See Theorem 5.2.3)Let S = A [x 1 , ••• , x 1 ] be a graded algebra over a Noetherian local ring (A, m) ⊂ S 0 . Let ϕ = {ϕ(n)} n≥0 be a I -good filtration of ideals ϕ(n) of R and ϕ(1) = ( f 1 , f 2 , ..., f r ) with deg f i = d i be Z-homogenous ideal in S, and let R = S[T 1 ,...,T n ] be a bigraded polynomial extension of S with deg(T i ) = (d i , 1) and deg(a) = (deg(a), 0) ∈ Z × {0} for all a ∈ S.

  of the Hilbert function HF (B,(b 1 , b 2 )).The problem is that P (b) is not full dimensional. To be able use the Barvinok algorithm, one should transform P (b) to the full dimensional polytope Q which has the same lattice point as P (b) .The following lemma gives us the complete information about Q: Lemma 1.0.3. (See Lemma 3.5.4)Let A =

  then Rees algebra can be described as subring of the graded polynomial ring R [t ] and denoted by R [I t] . One can define the R-algebra surjective homomorphism as follows :

2. 1 . 5 . 2 Ratliff-Rush closure Definition 2 . 1 . 19 .:= n 1 ITheorem 2 . 1 . 20 .Lemma 2 . 1 . 21 .

 1522119121202121 Let A be a Noetherian local ring and I ⊂ A an ideal with grade(I ) > 0 . The Ratliff-Rush closure of I is the ideal:I n+1 : I n .One of basic facts about Ratliff-Rush closure of powers of ideals is the following: Let I be an ideal containing regular elements. Then there exists an integer n 0 such that I n = I n for n n 0 .So we can define an invariant for I ρ(I ) := mi n{n 0 0| I n = I n f or al l n n 0 } and it can be calculated by the following lemma. Let (A, m) be a local Noetherian ring and let I ⊂ A be an ideal of grade(I ) > 0.

Definition 3 . 1 . 1 .

 311 Let G be an abelian group. A G-grading of S is a group homomorphism deg :Z n -→ G and deg(x u ) := deg(u) for a monomial x u = x u 1 1 ...x u n n ∈ S. An element c u x u ∈ S is homogeneous of degree µ ∈ G if deg(u) = µwhenever c u = 0 and an ideal I ⊂ S is homogeneous if every polynomial in I is a sum of homogeneous polynomials under the given grading. Theorem 3.1.2. (See [37, Theorem 8.6]) Let Q = deg(N n ) and L = ker(deg) of the above group homomorphism . Then the following canditions are equivalent for a polynomial ring S graded by G.

Definition 3 . 1 . 3 .Remark 3 . 1 . 4 .

 313314 [START_REF] Miller | Combinatorial commutative algebra[END_REF] If the equivalent conditions of the above theorem hold for a torsion-free abelian group G then we call grading by G positive, and for the polynomial ring S = k[x 1 ,..., x n ] which is graded by G we say that S is a positively graded polynomial ring. There is a two important cases about the image of group homomorphism deg :Z n -→ Gwhere it has torsion or not. In the case where image of deg has a torsion it can be happen that associated primes of G-graded S-module M are not graded but in the other the associated primes of M are graded by following proposition. Proposition 3.1.5. (See[START_REF] Miller | Combinatorial commutative algebra[END_REF] Theorem 8.11]) Let S = k[x 1 ,..., x n ] be a polynomial ring over field k graded by a torsion-free abelian group G and let M be a G-graded S-module. If P be an associated prime of M then P is homogeneous and P = ann(m) where m is homogeneous element of M .

Definition 3 . 2 . 4 .Remark 3 . 2 . 5 . 1 .

 3243251 and relint A := { j ∈C λ j p j |λ j > 0, ∀ j ∈ C } The above definitions help us to understand the following definition of polyhedral subdivision of a set of vectors. [22] A polyhedral subdivision of a vector configuration A is a collection ℜ of subconfigurations of A in R d that satisfies the following conditions : 1. If C ∈ ℜ then all the faces of C are also in ℜ , 2. C ∈ℜ Cone(C ) ⊇ Cone(A) , 3. relint(C ) ∩ relint(C ) = for C ,C ∈ ℜ implies that C = C The elements of a polyhedral subdivision ℜ are called cells. Cells of the same rank as A are maximal. Cells of rank 1 are called rays of ℜ. A triangulation of A is a polyhedral subdivision whose cells are simplices.

1 Figure 3 . 1 : 2 a 5 a 4 a 3 a 1 Figure 3 . 2 :Theorem 3 . 2 . 7 .

 1312543132327 Figure 3.1: Triangulation of pentagonal cone for a 2-dimensional slice

Definition 3 . 3 . 4 .Remark 3 . 3 . 5 .

 334335 For a set A ⊂ R d , the indicator function of A defined by [A] : R d → R Write the vector space ℑ(R d ) over Q generated by the indicator functions of all polyhedra inside R d also ℑ c (R d ) ( ℑ k (R d ) )denote for the subspace spanned by the indicator functions of polytopes (respectively. cones ) in R d . The pointwise multiplication of indicator functions make's ℑ(R d ) a commutative algebra and ℑ c (R d ),ℑ k (R d ) are subalgebra of ℑ(R d ). Definition 3.3.6. A linear transformation Ψ : ℑ(R d ) → V where V is a vector space over Q, is called valuation.

Figure 3 . 3 :

 33 Figure 3.3: Unimodular decomposition.

Example 3 . 3 . 14 .

 3314 Let r = -42 10 then by the above we write write the expansion of r as [-5; 1, 3, 1].

[Figure 3 . 4 :

 34 Figure 3.4: changing into the cone generated by (1, 0) and some other integer vector

Proposition 3 . 4 . 2 .

 342 a 2 × n-matrix with entries in N such that d 1 ≤ ... ≤ d n . Then the chambers corresponding to Pos(A) are positive polyhedral cones (∆) where ∆ is generated by {(d i ,1),(d i +1 , 1)} for all d i = d i +1 where i runs over {1,...,n}. Proof. Since any arbitrary pair {(d i , 1), (d j , 1)} makes an independent set whenever d i = d j , the common refinement consists of disjoint union of open convex polyhedral cones generated by {(d i , 1), (d i +1 , 1)} for all i = 1,...,n s.t.d i = d i +1 . Now we are ready to prove the main result of this section. Let B = k[T 1 ,...,T n ] be a bigraded polynomial ring over field k with deg(T i ) = (d i , 1).

Example 3 . 4 . 3 .

 343 Let B = k[T 1 , T 2 , T 3 ]be a polynomial ring over field k and deg

Theorem 3 . 5 . 3 .

 353 extracted from A with respect to which b has positive coordinares called b-positive. 2. A point c ∈ Pos(A) is called strongly regular if there is no sublist Y ⊂ A lying in a proper vector subspace , such that c ∈ Pos(Y ). A point in Pos(A) is called strongly singular if it is not strongly regular . For using the Barvinok algorithm on the variable polytope P (b) we need to know about it's structures as vertices and faces which done by the following theorem . [16] Let b ∈ Pos(A) be a strongly regular point . Then (i) The vertices of P (b) are of the form P Y (b) = {x ∈ R r |Y x = a; x 0} with Y a b-positive basis. (ii) The faces of P (b) are of the form P Z (b) where Z runs over the subsets of A containing a b-positive basis. |Z |d is the dimension of P Z (b) and positive basis in Z correspond to vertices of P Z (b)

Remark 3 . 5 . 5 .Theorem 3 . 5 . 6 .

 355356 In the above polytope Q when we fix a chamber there are some vertices which are inactive, more precisely if the point b = (b 1 , b 2 ) is in the first chamber then b 1d 1 b 2 0 and d 2 b 2b 1 0 so only vertices Q 6 ,Q 5 ,Q 4 are active. Now we are able to optain the explicit formula for non-standard Hilbert function by above lemma. Let B = k[T 1 ,...,T 4 ] be a graded polynomial ring over a field k with deg T i = (d i , 1) and

.

  If b = (b 1 , b 2 ) ∈ C 1 then it is clear from 3.5.4 that HF (B,(b 1 , b 2 )) = f C 1 (Q, (1,1)) however f has a pole at x = (1, 1) , it is analytic at x = (1, 1). Because of cancelation of the coefficients of negative powers in the Luarent series at x = (1, 1) so the value at x = (1, 1) it is the sum of the cofficients of the constant term in Laurent series of each term in 3.5.4[START_REF] Barvinok | A course in convexity[END_REF], to be able of computions first we change f from multivariate to univariate by following the Yoshida at al.[START_REF] De Loera | Effective lattice point counting in rational convex polytopes[END_REF] method.

Theorem 4 . 1 . 2 .

 412 Let R be a Noetherian local ring with maximal ideal m. Let M be a finitely generated R-module and M = n 0 I n M . Then both pr R ( M I n M ) and i d R ( M I n M ) stabilizes for k 0.

Example 4 . 1 . 3 .

 413 [START_REF] Herzog | The Betti polynomials of powers of an ideal[END_REF] Let I = (x 3 , x 2y z, y 4 + xz 3 , x yz 2 ) ⊂ R = k x, y, z , the Kodiyalam polynomials of I are as follows:

Proposition 4 . 2 . 2 .

 422 With notations as above, let B be a finitely generated Z d -graded R-module. There exist convex sets of dimension d in R d of the form

Lemma 4 . 3 . 1 .

 431 For a strictly increasing sequence d 1 < ••• < d r , and points of coordinates (β j 1 ,

Figure 4 . 2 :

 42 Figure 4.2: regions when n is sufficiently large.

  of Tor module of Rees algebra Let S = A[x 1 ,..., x n ] be a graded algebra over a commutative noetherian local ring S 0 = (A, m) with residue field k and set R = S[T 1 ,...,T r ] and B = k[T 1 ,...,T r ] . We set deg(T i ) = (d i , 1) and extended the grading from S to R by setting deg(x i ) = (deg(x i ), 0). In this section we use the two important following fact that were already at the center of the work [6] . The first one is that Tor R i (M R I , B ) is a finitely generated graded B -module. The second is that : Tor R i (M R I , B ) (µ,t ) = Tor S i (M I t , k) µ . 71

Theorem 5 . 1 . 1 .Theorem 5 . 1 . 2 .

 511512 [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] Let S = A[x 1 ,..., x n ] be a G-graded algebra over Notherian local ring (A, m, k). LetI = ( f 1 , f 2 , ..., f r ) with d eg f i = d i be G-homogenous ideal in S, and let R = S[T 1 ,...,T n ] be a bigraded polynomial extension of S with deg(T i ) = (d i , 1) and deg(a) = (deg(a), 0) ∈ G × {0} for all a ∈ S. Let M be a finitely generated G-graded S-module. Then for all i , j :1. Tor A i (Tor R j (M R I , A), k) is a finitely generated k[T 1 ,...,T r ]-module .2. Tor R i (M R I , k) is a finitely generated k[T 1 ,...,T r ]-module . In the above situation if I is a homogeneous ideal in S and G = Z.

2 PRELIMINARIES 2.1 commutative algebra 2.1.1 Graded Rings and Modules Definition 2.1.1

. A N-graded ring R is a ring together with a decomposition (as abelian groups

  1.2. A graded R-module is an ordinary R-module M together with a decomposition M = ⊕ k∈Z M k as abelian groups, such that R m .M n ⊆ M m+n for all m ∈ Z ≥0 and n ∈ Z. Elements in one of these pieces are called homogeneous. Any m ∈ M is thus uniquely a finite sum m n i where each m n i ∈ M n i is homogeneous of degree n i and n

µ . Remark 2.1.15. By

	Proposition 2.1.16. Let S = k[x 1 ,..., x n ] be a positively graded Z d -graded polynomial ring over the field
	k. Then the followings hold:
	1. The Hilbert series of S(-µ) is the development in Laurent series of the rational function
	[37, 8.8], if S is positively graded by Z d , then the semigoup Q = deg(N n ) can be em-
	bedded in N d . Hence, after such a change of embedding, the above Hilbert series are Laurent series in
	the usual sense.
	We recall that the support of a Z d -graded module N is
	Supp Z d (N ) := {µ ∈ Z d |N µ = 0},
	and use the abbreviated notations Z[t ] := Z[t 1 ,..., t d ] for t = (t 1 ,..., t d ) and t µ := t	µ 1 1 ••• t	µ d d for µ = (µ 1 ,...,µ d ) ∈
	Z d .

  ⊂ R d be a polytope where its vertices have a rational coordinate. Let nP = {nx|x ∈ P } for a positive integer n then the function #(nP ∩Z d ) is a quasi-polynomial of degree dim P . Further more if C is an integer where C P is an integer polytope, then C is a period of #(nP ∩ Z d ) . In the particular case where P is an integer polytope , the Ehrhart polynomial is as follows :

	One of important generalizations of the Pick formula is the theorem of Ehrhart which shows the
	polynomial behavior of lattice point enumeration independ of the dimension.
	Theorem 3.3.3. [Ehrhart 1977] Let P
	First we recall the definition of a polytope and a very classical of G.Pick[1899] for a two-dimensional
	polygone :
	Proposition 3.3.2. [G.Pick] Suppose that P ⊂ R 2 is an integer polygon. Then the number of integer points
	inside P is :
	|P ∩ Z 2 | = ar ea(P ) +	|∂P ∩ Z 2 | 2	+ 1

rational polyhedron is called a polytope. A polytope P ⊂ R d is called integer polytope if its vertices are points from Z d

  d j and suppose that gcd(Y 12 , Y 13 , Y 23 ) = 1. Then there exist f i j , g i j such that f 12 Y 23 + g 12 Y 23 = gcd(Y 23 , Y 13 ) gcd( f 12 Y 13 + g 12 Y 23 , Y 12 ) = 1, f 13 Y 12 + g 13 Y 23 = gcd(Y 12 , Y 23 ) gcd( f 13 Y 12 + g 13 Y 23 , Y 13 ) = 1, f 23 Y 13 + g 23 Y 12 = gcd(Y 13 , Y 12 ) gcd( f 23 Y 13 + g 23 Y 12 , Y 23 ) = 1, with f 12 Y 13 + g 12 Y 23 -1 f 12 Y 13 + g 12 Y 23 ≡ 1 mod Y 12 f 13 Y 12 + g 13 Y 23 -1 f 13 Y 12 + g 13 Y 23 ≡ 1 mod Y 13

f 23 Y 13 + g 23 Y 12 -1 f 23 Y 13 + g 23 Y 12 ≡ 1 mod Y 23 .

and f 12 , g 12 , f 13 , g 13 , f 23 and g 23 can be calculated by an Euclidean algorithm.

Our chambers are regions

  Theorem 4.2.1. Let R = S[T 1 ,...,T r ] be a G ×G -graded polynomial extension of S with deg G×G (a) ∈ G ×0for all a ∈ S and deg G×G (T × Z s )-graded ring, over k[T i , j ] setting deg(T i , j ) = (deg( f i , j ), e i ) with e i the i -th canonical generator of Z s and f i , j 's minimal generators of I i . Hence, Tor R i

	is a finitely generated (Z p
			dim k Tor A j (Tor S i (M ( * ,t ) , A) δ , k)
	is polynomial in the t i s for t	0, and the function
			dim k Tor S i (M ( * ,t ) , k) δ
	is polynomial in the t i s for t	0.
	Now we return to the main result on Betti numbers of powers of ideals. We can treat without any fur-
	ther effort the case of a collection of graded ideals and include a graded module M . Hence, we will study
	the behaviour of dim k Tor R i (M I	t 1 1 ••• I	t s s , k) µ for µ ∈ Z p and t	0. To this aim, we first use the important
	fact that the module
		B t 1 1 ••• I	t s s , k)

j ) ∈ 0 × G for all j . Let M be a finitely generated G × G -graded R-module and let i be an integer. Assume that i = 0 or A is a Noetherian ring. Then 1. There exists a finite subset ∆ i ⊆ G such that, for any t , Tor S i (M ( * ,t ) , A) δ = 0 for all δ ∈ ∆ i .

2. Assume that G

= Z s . For δ ∈ ∆ i , Tor S i (M ( * ,t ) , A) δ = 0 for t 0 or Tor S i (M ( * ,t ) , A) δ = 0 for t 0. If,

furthermore, A → k is a ring homomorphism to a field k, then for any j , the function i := ⊕ t 1 ,...,t p Tor R i (M I

  let E be the set of d -tuples e = (e i 1 , ..., e i d ) such that (ϕ(e i 1 ), ..., ϕ(e i d )) generates a lattice Λ e in Z d , and set

  [START_REF] De Concini | Topics in hyperplane arrangements, polytopes and boxsplines[END_REF], there exists a polynomial κ B (t 1 ,..., t d ) with integral coefficients such that B (t ) = a∈A c a t a with A ⊂ ∪ Supp Z d (Tor R (B, k)). Let D i := ∪ j {x | H i , j (x) = 0} be the minimal union of hyperplanes containing the border of C i . The union C of the convex sets C i + a can be decomposed into a finite union of convex sets ∆ u , each u ∈ U corresponding to one connected component of C \ i ,a (D i + a). (Notice that R d \ i ,a (D i + a) has finitely many connected components, which are convex sets of the form of ∆ u , and that we may drop the ones not contained in C as the dimension of B ξ is zero for ξ not contained in any C i + a.) We define u as the set of pairs (i , a) such that (C i + a) ∆

H (B ; t ) = κ B (t )H (R; t )

and κ u = , and

remark that if (i , a) ∈ u then ( j , a) ∈ u for j = i . If ξ ∈ i C i + a, then dim k R ξ-a = 0, while if ξ ∈ C i + a then it

follows from Corollary 3.2.8 that there exist polynomials Q

where i ∈ {1, -1}, a i ∈ Z d and b i 1 , ••• , b i d form a basis of Z d for each i . Suppose that the vectors c i for 1 i m are fixed and the β i vary in such a way that the combinatorial structure of polyhedron P = P (β) stays the same . Then the exponents b i j in the denominators remain the same , whereas the exponents a i = a i (β) in the numerator change with β ∈ Q m as

Where the l i j : Q m :→ Q are linear functions. If β is such that P (β) is an integer polytope, then l i j ∈ Z for each pair i,j . using the 3.3.13 and 3.3.16 we can decompose the generating function of Cone(Q, q 5 ) f (Cone(Q, q 5 ), (x 1 ,

we can decompose Cone(Q, q 4 ) in the same way [Cone(Q,

So ( 3) can be achieved from 3.3.13.

ASSYMPTOTIC BEHAVIOR BETTI NUMBER OF POWERS

OF IDEALS

Kodiyalam Polynomials

Let R be a Noetherian local ring with maximal ideal m and residue field k. Let I be a proper ideal of R. Kodiyalam in [START_REF] Kodiyalam | Homological invariants of powers of an ideal[END_REF] proved the polynomial behavior of Betti number and Bass number of a finitely generated graded module as follows : Denote as above by Λ the lattice in Z d associated to ϕ, by s Λ : Z d → Z d /Λ the canonical morphism and by C i , for i ∈ F , the maximal cells in the chamber complex associated to ϕ. One has

Theorem 4.2.4. In the situation above, there exist a finite number of polyhedral convex cones

Furthermore, for any

Proof. It has been presented in [START_REF] Bagheri | The eventual shape of the Betti tables of powers of ideals[END_REF] This general finiteness statement may lead to pretty complex decompositions in general, that depend on the number of ideals and on arithmetic properties of the sets of degrees of generators. This complexity is reflected both by the covolume of Λ as defined above and by the number of simplicial chambers in the chamber complex associated to ϕ.

Now we are ready to give a specific description of Tor S i (I t , k) in the case of a Z-graded ideal. Let E := {e 1 ,...,e s } with e 1 < ••• < e s be a set of positive integers. For from 1 to s -1, let

be the closed cone spanned by e 1 and e +1 1 . For integers i = j , let Λ i , j be the lattice spanned by e i 1 and e j 1 and

Proposition 4.3.2. In the above situation, if M is a finitely generated graded B -module, there exist t 0 , N and L i (t ) := a i t + b i for i = 1,..., N with b i ∈ Z and {a 1 ,..., a N } = E such that for t ≥ t 0 :

Now let L j i (t ) := d i t +b j be the half-line parallel to the vector (d i , 1) and passing through the point (β

for 1 ≤ i , j ≤ N . Then by description before proposition item(i) follows directly from 4.3.1 (i) and item(ii)

To prove (iii), following 4.3.1 we can consider two type of intervals as follows:

and

And wite I N p +q := I σ p (q) p ,L N p +q = L σ p (q) p for 0 ≤ q < N then for any degree (α, n) in the support of M there is two cases : case I. if α ∈ I j i , then (α, n) belongs to i-th chamber of shifts {(β

),...,(β

,

)} and for the other shifts (α, n) belongs to (i -1)-th chambers.

case II. if α ∈ I N i , then (α, n) belongs to i -th chamber for all of the shifts.

Then by Proposition 3.4.2 there exist polynomials Q i j such that dim B (µ,t ) = Q i j ((µ, t )), if µtd i ≡ j mod (∆).

By setting

we can conclud that: (5) Let γ be I -good filtration , then a J -good filtration ϕ is called good reduction of γ if it is a reduction in the sense of (3).

In opposite to the ideal case, minimal reduction of a filtration does not exist in general. But Hoa and Zarzuela showed in [START_REF] Hoa | Reduction number and a-invariant of good filtrations[END_REF] the existence of a minimal reduction for I -good filtrations as follows :

Proposition 5.2.2. Let ϕ and γ are filtrations on R, then ϕ is the minimal reduction of a good filtration γ if and only if ϕ = {J n } n 0 , where J is a minimal reduction of γ 1 ). In particular, a minimal reduction of γ do exist.

that is why we are interested about I -good filtration to generalize the previous results. The following theorem explain the structure of Tor module of I -good filtrations: (1)Then for all i :

Tor R i (R ϕ , k) is a finitely generated k[T 1 ,...,T r ]-module.


Didier Dubois

Fuzzy weighted averages and fuzzy convex sums: Author's response

Keywords: Interval analysis, Weighted average, Convex sums

We elaborate further on the reasons for the lack of equality between interval weighted averages and interval-based convex sums and highlight its consequences on the cogency of multicriteria decision evaluation procedures based on the use of fuzzy interval weights.

In his note, Pavlačka [START_REF] Pavlačka | Note on the lack of equality between fuzzy weighted average and fuzzy convex sum[END_REF] corrects a technical mistake appearing in my position paper [START_REF] Dubois | The role of fuzzy sets in decision sciences: old techniques and new directions[END_REF]. I am grateful for this correction for two reasons:

• The geometrical analysis of constraining ill-known normalized vectors vs. normalizing ill-known weights is very instructive on the difference between the two methods. • Moreover, this result seems to raise an interesting issue in the use of interval and fuzzy extensions of weighted averages in decision analysis.

From a geometrical point of view, the problem is to study the relative positions of the sets of probability vectors

W = w 1 n i=1 w i , . . . , w n n i=1 w i : w i ∈ [a i , b i ], i = 1, . . . , n
and

W N = ( p 1 , . . . , p n ) : p i ∈ a i a i + j i b j , b i b i + j i a j , i = 1, . . . , n, n i=1 p i = 1 .
For n = 2 these sets are equal. Consider the case n > 2 and for simplicity we assume a i > 0, ∀i. It can be observed that the polyhedra W and W N both lie in the hyperplane H = { p = ( p 1 , . . . , p n ) : n i=1 p i = 1} of normalized vectors, but no facet of one is parallel to a facet of the other. Let P = { p ∈ H : p i Ն 0, i = 1, . . . , n} be the positive part of H, that is, the set of probability assignments in H.

It is clear that W N is the intersection between P and the Cartesian product defined by × i=1,. ..,n [a i /(a i + j i b j ), b i /(b i + j i a j )]. Each pair of parallel facets of W N of the form p i = p, where p = a i /(a i + j i b j ) and p = b i /(b i + j i a j ) is also parallel to the facet of P such that p i = 0, and this for i = 1, . . . , n. In contrast, W is obtained by homothetic projection, with center equal to the origin (0, . . . , 0), of the Cartesian product × i=1,. ..,n [a i , b i ] on P. The vertices of this polyhedron lie inside the facets of W N as shown in [START_REF] Pavlačka | Note on the lack of equality between fuzzy weighted average and fuzzy convex sum[END_REF], so that both W N and W have the same marginal projections of the form [a i /(a i + j i b j ), b i /(b i + j i a j )]. Moreover, each edge of W is the homothetic projection of one edge of the hyper-rectangle × i=1,. ..,n [a i , b i ] and lies in the same hyperplane as one axis of the referential, hence on a line containing the vertex of P on that axis. The set W is defined via constraints of the form p i / p j = w i /w j , while W N is generated from bounds on individual p i . This type of representation is common in the area of imprecise probabilities [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF], where credal sets of the form W N are obtained by constraints on the probabilities of elementary events [START_REF] De Campos | Probability intervals: a tool for uncertain reasoning[END_REF] while those of the form W are obtained by constraining conditional probabilities.

For instance consider the case where n = 3 pictured by Fig. 1 in Pavlačka [START_REF] Pavlačka | Note on the lack of equality between fuzzy weighted average and fuzzy convex sum[END_REF], where P is an equilateral triangle with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1). The set W of renormalized weights is a hexagon whose sides are obtained by lines drawn from the vertices of the triangle (two per vertex). In contrast, W N is the smallest hexagon, with opposite edges parallel to the edges of the triangle P, containing W. In the fuzzy case, we get fuzzy subsets of P consisting of nested hexagons.

It should be clear from [START_REF] Pavlačka | Note on the lack of equality between fuzzy weighted average and fuzzy convex sum[END_REF] and the above additional comments that the vertices of W are generally not the same as the vertices of W N . As a consequence, the intervals

X = n i=1 x i • w i n i=1 w i : w i ∈ [a i , b i ], i = 1, . . . , n and X N = n i=1 x i • p i : p i ∈ a i a i + j i b j , b i b i + j i a j , i = 1, . . . , n, n i=1 p i = 1
have little chance to be equal because their bounds are attained on vertices of W and W N respectively. And as W ⊂ W N it follows that, in general, X ⊂ X N . A consequence of this finding is that in multicriteria decision-making based on fuzzy weighted averages, constraining ill-known normalized vectors and normalizing ill-known weights will give different results. Note that in the precise case, one may indifferently compute a convex sum of ratings according to the various criteria either using normalized weight vectors or using a weighted sum with suitable non-normalized weight vectors. As the results without normalization will be proportional to the results using a normalized vector, this choice of method is immaterial for determining the ranking of available options. However in the interval-valued case (let alone in the fuzzy case) there is no guarantee that the rankings of options will be the same with the two approaches, respectively using intervals X and X N for comparing the options, as they rely on imprecision polyhedra having different vertices. All we know is that X will be strictly included in X N . Even worse, there is generally no way of having the polyhedron of normalized vectors induced by unnormalized interval weights equal to a polyhedron induced by suitable ranges of imprecision bearing on components of a normalized weight vector, due to the general incompatibility of their respective configurations.

This dilemma is very problematic when applying fuzzy weighted averages to multicriteria decision analysis. Namely, how to model the imprecision on the weights bearing on each criterion? Usually a weighted average is defined as a convex sum of ratings. Should we elicit imprecision on normalized weight vectors and compute a fuzzy convex sum? or elicit imprecision on unnormalized weights and compute the image of the fuzzy weights via a rational fraction as done by most authors since Baas and Kwakernaak [START_REF] Baas | Rating and ranking of multiple aspect alternatives using fuzzy sets[END_REF] proposed it? The useful correction of Pavlačka [START_REF] Pavlačka | Note on the lack of equality between fuzzy weighted average and fuzzy convex sum[END_REF] should lead to a reappraisal of the cogency of the fuzzy weighted average approach to decision analysis.

The natural way out of this dilemma may come from the following considerations. 1 Note that the two intervals X 1 and X 2 (likewise X N 1 and X N 2 ) evaluating two options 1 and 2 do not constrain independent quantities: Each assignment of weights w i in [a i , b i ] determines single values, one in X 1 and one in X 2 , that are linked to each other. So the final ranking of options based on (fuzzy) interval global ratings cannot be achieved by applying any (fuzzy) interval 1 Based on a suggestion by Eyke Huellermeier, gratefully acknowledged. ranking method to one of the sets of ill-known ratings X k or X N k . The interval ratings obtained for each option are not independent. A natural ranking criterion is then the following: If one option is better than another one for each assignment of weights in their ranges, then the former option is better than the latter. Note that this ranking criterion yields results that the renormalization procedure will leave unchanged: with this criterion, comparing X 1 and X 2 , or X N 1 and X N 2 , would give the same results. As pointed out by a reviewer of this note, this fact pleads in favor of extending, to intervals or fuzzy intervals, the whole decision procedure at once, including the aggregation and the ranking steps, so as to take the interactivity between the fuzzy global evaluation intervals into account.

E-mail address: dubois@irit.fr.