Marcel Morales

Dung Thai Nguyen

Thi Hilbert

Segre Transform

Castelnuovo-Mumford Regularity

Nguyen Thi

Hilbert series of Segre transform, and Castelnuovo-Mumford regularity

Keywords: Segre-Veronese, Castelnuovo-Mumford regularity, Cohen-Macaulay, postulation number 13D40, Secondary 14M25, 13C14, 14M05

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Introduction

In this paper, we deal only with formal Laurent series a = l σa

a l t l , σ a ∈ Z, a l ∈ C such that ( * ) a = h(a)(t) (1 -t) da , for some d a 0, h(a)(t) ∈ C[t, t -1 ].
Given two formal Laurent series a, b satisfying (*), the Segre transform a⊗b is defined by

a⊗b = l σ a l b l t l ,
1 Partially supported by VIASM, Hanoi, Vietnam.

where σ = max{σ a , σ b }. In a recent preprint [F-K], the authors proved the bilinearity of the Segre transform under some restricted hypothesis, motivated by this results we show in this paper the bilinearity of the Segre transform in general. We apply these results to compute the postulation number of a Segre product of series satisfying (*). Property (*) is equivalent to the existence of a polynomial Φ a (t) ∈ C [t], such that a n = Φ a (n) for n large enough. The postulation number is the smallest integer β a such that a n = Φ(n) for n > β a . It is well known that β a = deg h(a)(t)d a .

Our second application is motivated by the paper [C-M], where is computed the Castelnuovo-Mumford regularity of the Segre Veronese embedding, we can extend partially their result. Our main result is:

Theorem. Let S 1 , . . . , S s be graded polynomial rings on disjoints of set of variables. For all i = 1, . . . , s, let M i be a graded finitely generated S i -Cohen-Macaulay module. We assume that

M i = ⊕ l 0 M i,l as S i -module. Let d i = dim M i , b i = d i -1 ≥ 0, α i = d i -reg(M i ), where reg(M i ) is the Castelnuovo-Mumford regularity of M i . If reg(M i ) < d i , for all i = 1, . . . , s then (1) M 1 ⊗ . . . ⊗M s is a Cohen-Macaulay S 1 ⊗ . . . ⊗S s -module. (2) reg(M 1 ⊗ . . . ⊗M s ) = (b 1 + . . . + b s + 1) -max{α 1 , . . . α s }. (3) For n i ∈ N, let M <n i > i be the n i -Veronese transform of M i , then reg(M <n 1 > 1 ⊗ . . . ⊗M <ns> s ) = (b 1 + . . . + b s + 1) -max{⌈ α 1 n 1 ⌉, . . . , ⌈ α s n s ⌉}.
Note that this result can be proved easily by using local cohomology, but our purpose is to give a very elementary proof.

Segre transform of Laurent series

In this paper, we deal only with formal Laurent series

a = l σa a l t l , σ a ∈ Z, a l ∈ C such that ( * ) a = h(a)(t) (1 -t) da , for some d a 0, h(a)(t) ∈ C[t, t -1 ].
We will set h(a)(t) = n σa h n (a)t n .

Definition 1.1. Let a, b be two formal Laurent series satisfying (*). the Segre transform a⊗b is defined by

a⊗b = l σ a l b l t l ,
where σ = max{σ a , σ b }.

In all this paper we assume that a⊗b = 0.

Lemma 1.2. a⊗b satisfies ( * ).

Proof. By [M1], property (*) is equivalent to the existence of a polynomial Φ a (l) such that Φ a (l) = a l for l large enough. Moreover,

d a = deg Φ a + 1 if Φ a is a non zero polynomial d a = 0 if Φ a = 0.
We have also a polynomial Φ b (l) such that Φ b (l) = b l for l large enough. Hence a l b l = Φ a (l)Φ b (l) is a polynomial for l large enough, and again by [M1], there exist a Laurent polynomial h(a⊗b)(t) such that

a⊗b = h(a⊗b)(t) (1 -t) d a⊗b ,
where

d a⊗b = 0 if either d a = 0 or d b = 0 d a + d b -1 if d a , d b 1.
Remark 1.3. We recall that binomial coefficients can be defined in a more general setting than natural numbers, indeed for k ∈ N, binomial coefficients are polynomial functions in the variable n. More precisely:

(1)

If k = 0 then let n 0 = 1, for all n ∈ C. (2) If k > 0 then let n k = n(n-1)...(n-k+1) k!
, for all n ∈ C.

Note that for all n ∈ C,

n k = (-1) k k-n-1 k and if n ∈ N, n < k, then n k = 0.
2 Segre transform is bilinear

Let recall the following Lemma 1 from [M1].

Lemma 2.1. Let a = l σa a l t l = h(a)(t) (1 -t) da , with h(a)(t) = h σa t σa + . . . + h ra t ra , d a 0 and σ a ≤ r a ∈ Z. We will set b a = d a -1.
Then for all n = σ a , . . . , r a we have

h n (a) = n-σa k=0 (-1) k d a k a n-k = n k=σa (-1) n-k d a n -k a k . (1)
On the other hand we have for all k σ a

a k = k-σa i=0 h k-i (a) b a + i i = k i=σa h i (a) b a + k -i k -i (2)
The first claim follows from the equality:

( l σa a l t l )((1 -t) da ) = h(a)(t).
The second claim since:

l σa a l t l = h(a)(t) (1 -t) da = (h(a)(t))( i 0 b a + i i t i )
The following two theorems extend [F-K, Theorem 1].

Theorem 2.2. Let a, b be two formal power series satisfying property (*). If

d a = 0 , then for all n ∈ Z h n (a⊗b) = a n b n = h n (a) n j=σ b h j (b) b b + n -j n -j . Moreover, deg h(a⊗b)(t) deg a. If d b > 0 and h j (b) 0, for all j then deg h(a⊗b)(t) = deg(a).
Proof. Since d a = 0, a is a Laurent polynomial, we have that h n (a) = a n , for all n ∈ Z and a n = 0 for n > deg(a), which implies that

a⊗b = n deg(a)
a n b n t n .

hence deg h(a⊗b)(t) ≤ deg(a). Now suppose that d b > 0 and h j (b) 0 for all j, by using equation (2) we have,

h n (a⊗b) = h n (a) n j=σ b h j (b) b b + n -j n -j . Note that h σ b > 0, b b +n-σ b n-σ b > 0, for n ≥ σ b . The assumption (a⊗b) = 0 implies σ b deg a, so deg a j=σ b h j (b) b b +deg a-j deg a-j > 0.
Hence h deg a (a⊗b) > 0 and the claim is over.

Remark 2.3. Let a, b be two formal power Laurent series satisfying property (*), the product h i (a)h j (b) is null for any i < σ, j < σ, where σ is any of the numbers σ a , σ b , σ (a⊗b) = max(σ a , σ b ), min(σ a , σ b ). Then for any n ∈ Z

h n (a⊗b) = ∞ i=σ ∞ j=σ h i (a)h j (b) b a + j -i n -i b b + i -j n -j .
The proof will follow immediately from the Lemma 2.1 and the proof of [F-K, Theorem 1]:

3 Postulation number, Castelnuovo-Mummford regularity Lemma 3.1. Let i, j ∈ Z, b 1 := d 1 -1 0, b 2 := d 2 -1 0 and T i d 1 := t i (1 -t) d 1 ; T j d 2 := t j (1 -t) d 2 Then T i d 1 ⊗T j d 2 = r i,j n=max(i,j) b 1 +j-i n-i b 2 +i-j n-j t n (1 -t) b 1 +b 2 +1 , (3)
where :

r i,j = min(b 1 + j, b 2 + i) if b 2 + i -j 0 and b 1 + j -i 0 max(b 1 + j, b 2 + i) if b 2 + i -j < 0 or b 1 + j -i < 0.
Proof. The equality (3) follows from theorem 2.4. We need to check that for n > r i,j , we have b 1 +j-i n-i b 2 +i-j n-j = 0, and that for n = r i,j , we have b 1 +j-i n-i b 2 +i-j n-j = 0. We have two cases:

1. If b 1 + j -i ≥ 0 and b 2 + i -j ≥ 0, then b 1 +j-i n-i = 0 if and only if b 1 + j < n. Hence r i,j = min(b 1 + j, b 2 + i).

Either b

1 + j -i < 0 or b 2 + i -j < 0. Suppose for example that b 1 + j -i < 0 then b 1 +j-i n-i = 0, and b 2 +i-j n-j = 0 if and only if b 2 + i < n, but b 1 + j < i ≤ b 2 + i < n. Hence r i,j = max(b 1 + j, b 2 + i). Example 3.2. Let α, β ∈ Z, we study T α d ⊗T β 1 . We consider two cases: (1) If max(α, β) = α then T α d ⊗T β 1 = t α (1 -t) d . (2) If max(α, β) = β > α then T α d ⊗T β 1 = t α -t α ( β-α-1 l=0 d-1+l l t l )(1 -t) d (1 -t) d . Note that deg(t α -t α ( β-α-1 l=0 d-1+l l t l )(1 -t) d ) = d -1 + β. Proposition 3.3. Let a = h(a)(t) (1 -t) da ; b = h(b)(t) (1 -t) d b , where h(a)(t), h(b)(t) ∈ C[t, t -1 ].
For any non null Laurent series satisfying property (*), we denote σ a = min n h n (a) = 0, r a = deg h(a)(t). Then (2) Since for all σ a i r a , σ b j r b , we have that b b + ij 0 and b a + ji 0, by the Lemma 3.1, we have that r i,j = min(b a + j, b b + i). This implies for all i, j that A i,j,r i,j = 0 and A i,j,n = 0 for n > r i,j .

On the other hand

min(b a + r b , b b + r a ) min(b a + r b , b b + i) min(b a + j, b b + i), for all σ a i r a , σ b j r b . Hence A i,j,n = 0 for n > min(b a + r b , b b + r a )
. Note that the conditions b a + ji 0, b b + ij 0 implies that A i,j,n 0 for all n. Hence if for all i, j, h i (a) 0, h j (b) 0 then h n (a⊗b) 0 for all n, and for m := min(b a + r b , b b + r a ) we have:

h m (a⊗b) = i,j|A i,j,m =0 h i (a)h j (b)A i,j,m > 0.
( (2) ⇒ (1). Take i = σ a , j = r b in the first inequality and j = σ b , i = r a in the second.

(1) ⇒ (2). Let σ a i r a , σ b j r b , then Proposition 3.7. Let consider a 1 , . . . , a s Laurent formal series satisfying (*)

i + b b -j i + b b -r b σ a + b b -r b 0 and j + b a -i i + b a -r a σ b + b a -r a 0.
a i = h(a i )(t) (1 -t) d i , h(a i )(t) ∈ C[t, t -1 ].
We set

r i = deg h(a i )(t), α i = d i -r i , b i = d i -1 0 and a 1 ⊗ . . . ⊗a s = h(a 1 ⊗ . . . ⊗a s )(t) (1 -t) b 1 +...+bs+1 . Then (1) deg(h(a 1 ⊗ . . . ⊗a s )) (b 1 + . . . + b s + 1) -min(α 1 , . . . , α s ).
(2) If the condition (**) of Lemma 3.5 is fulfilled for

{a 1 , a 2 }, {a 1 ⊗a 2 , a 3 }, . . . , {a 1 ⊗ . . . ⊗a s-1 , a s }
and h k (a i ) 0 for all i and k, then

deg(h(a 1 ⊗ . . . ⊗a s )) = (b 1 + . . . + b s + 1) -max(α 1 , . . . , α s ).
(3) If for all i = 1, ..., s, 0 ≤ σ i r i < d i , then the condition (**) of Lemma 3.5 is fulfilled for {a 1 , a 2 }, {a 1 ⊗a 2 , a 3 }, . . . , {a 1 ⊗ . . . ⊗a s-1 , a s }.

Proof.

(1) Note that

max(b 1 + r 2 , b 2 + r 1 ) = max(b 1 + b 2 + 1 -α 2 , b 1 + b 2 + 1 -α 1 ) = b 1 + b 2 + 1 -min(α 1 , α 2 ).
Now suppose s 3, we prove the claim by induction. Assume that (1) is true for the case s -1 : deg(h(a 1 ⊗ . . . ⊗a s-1 )) (b 1 + . . . + b s-1 + 1) -min(α 1 , . . . , α s-1 ).

Then, by the Proposition 3.3

deg(h(a 1 ⊗ . . . ⊗a s-1 )⊗a s )) max(b 1 + . . . + b s-1 + r s , b s + deg(a 1 ⊗ . . . ⊗a s-1 ) max(b 1 + . . . + b s-1 + b s + 1 -α s , b 1 + . . . + b s + 1 -min(α 1 , . . . , α s-1 ) = b 1 + . . . + b s + 1 + max(-α s , -min(α 1 , . . . , α s-1 )) = b 1 + . . . + b s + 1 -min(α s , min(α 1 , . . . , α s-1 )) = b 1 + . . . + b s + 1 -min(α s , min(α 1 , . . . , α s ))
(2) Since condition (**) is fulfilled, we can apply Proposition 3.3 and we get:

deg(a 1 ⊗a 2 ) min(b 1 + r 2 , b 2 + r 1 ) = min(b 1 + b 2 + 1 -α 2 , b 2 + b 1 + 1 -α 1 ) = b 1 + b 2 + 1 + min(-α 1 , -α 2 ) = b 1 + b 2 + 1 -max(α 1 , α 2 )
and we have equality if h k (a i ) 0, for all i, k. By induction hypothesis we assume that

deg(a 1 ⊗ . . . ⊗a s-1 ) (b 1 + . . . + b s-1 + 1 -max(α 1 , . . . , α s-1 ) (4)
and we have equality if h k (a i ) 0, for all i, k. Moreover by Proposition 3.3,the coefficients h k (a 1 ⊗ . . . ⊗a s-1 ) are 0. On the other hand, condition (**) is fulfilled, so we can apply Proposition 3.3, we have:

deg(a 1 ⊗ . . . ⊗a s-1 )⊗a s )) min(b 1 + . . . + b s-1 + r s , b s + deg(a 1 ⊗ . . . ⊗a s-1 ) (5),
where we have the equality if h k (a s ) 0 and h k (a 1 ⊗ . . . ⊗a s-1 ) 0, which is true since by hypothesis h k (a i ) 0 for all i, k.

Using (4) in (5) we get deg(h(a 1 ⊗ . . . ⊗a s )) min(b 1 + . . . + b s-1 + 1 -α s , b s + b 1 + . . . + b s-1 + 1 -max(α 1 , α s-1 )) = b 1 + . . . + b s + 1 + min(-α s , -max(α 1 , . . . , α s-1 )) = b 1 + . . . + b s + 1 -max(α s , max(α 1 , . . . , α s-1 )) = b 1 + . . . + b s + 1 -max(α s , min(α 1 , . . . , α s ))
and we have the equality if h k (a i ) 0 for all i, k.

(3) The proof is immediate from Proposition 3.3.

4 hvector of the Segre product of s power series

The proof of the following theorem is direct from 2.4 by using induction.

Theorem 4.1. With the notations of Proposition 3.7.

For

σ s ≤ i s ≤ b 1 + ... + b s -min{α 1 , ..., α s } we have h is (a 1 ⊗...⊗a s ) = (i 1 ,i 2 ,...,i s-1 ,l 2 ,...,ls)∈∆ h i 1 (a 1 )h l 2 (a 2 )...h ls (a s )A i 1 ,l 2 ,i 2 ...A i s-1 ,ls,is where ∀k = 2, ..., s; A i k-1 ,l k ,i k = b 1 + ... + b k-1 + l k -i k-1 i k -i k-1 b k + i k-1 -l k i k -l k ,
and ∆ is defined by : for any τ = 2, ..., s,

σ τ ≤ l τ ≤ b τ + 1 -α τ , σ τ -1 ≤ i τ -1 ≤ min{b 1 + ... + b τ -1 -min{α 1 , ..., α τ -1 }, i τ }.
There is an important corollary that will be used in [M1] to prove the conjecture by Simon Newcomb: Theorem 4.2. For j = 1, ..., n, let S j be a polynomial ring over a field K in b j + 1 variables, a j the Hilbert-Poincare series of S j , that is a j = 1 (1t) b j +1 then: Applying Proposition 3.7 to modules we get the following Theorem (note that this theorem can be proved easily by using [G-W], but our purpose is to prove it by using only elementary tools):

Theorem 4.3. Let S 1 , . . . , S s be graded polynomial rings on disjoints sets of variables. For all i = 1, . . . , s, let M i be a graded finitely generated S i -Cohen-Macaulay module. We assume that M i = ⊕ l 0 M i,l as S i -module. Let

d i = dim M i , b i = d i -1 ≥ 0, α i = d i -reg(M i ), where

  Theorem 2.4. Let a, b be any formal power Laurent series satisfying property (*). Let b a = d a -1 0, b b = d b -1 0 and σ be any of the numbers σ a , σ b , σ (a⊗b) = max(σ a , σ b ), min(σ a , σ b ).

( 1 )

 1 r a⊗b max(b a + r b , b b + r a ).(2) If for all σ a i r a , σ b j r b such that h i (a) = 0, h j (b) = 0 we have b b + ij 0 and b a + ji 0 then r a⊗b min(b a + r b , b b + r a ). Moreover, if for all i, j, h i (a) 0, h j (b) 0 then r a⊗b = min(b a + r b , b b + r a ). (3) If 0 σ a r a b a and 0 σ b r b b b , then r a⊗b min(b a + r b , b b + r a ). Moreover, if for all i, j, h i (a) 0, h j (b) 0 then r a⊗b = min(b a + r b , b b + r a ). Proof. (1) Let n > max(b a + r b , b b + r a ), for all i r a , j r b . By the Lemma 3.1, this implies that A i,j,n = 0, for all i r a , j r b . Hence h n (a⊗b) = 0, which implies that σ ( a⊗b) max(b a + j, b b + i).

  ) If 0 σ a r a b a and 0 σ b r b b b then b a + j b a i and b b + i b b j. Therefore b a + ji 0 and b b + ij 0. Hence, the claim follows from the claim 2. Remark 3.4. The bounds obtained are sharp. Lemma 3.5. The following statements are equivalent: (1) For all σ a i r a and σ b j r b , we have b b + ij 0 and b a + ji 0. (2) b b + σ ar b 0 and b a + σ br a 0. ( * * ) Proof.

  Remark 3.6. Suppose that M a , M b are Cohen-Macaulay modules of dimensions d a = b a +1 ≥ 2, d b = b b + 1 ≥ 2, with Hilbert-Poincaré series a, b. Then the conditions b b + σ ar b 0 and b a + σ br a 0 are equivalent to say that M a ⊗M b is a Cohen-Macaulay module by [G-W][Proposition (4.2.5)].

For

  k = 0, ..., b 1 + ... + b n -max{b 1 , ..., b n }, we haveA([b], k) = (i 2 ,...,i n-1 )∈∆ A i 2 A i 2 ,i 3 A i 3 ,i 4 ...A i n-1 ,in , where i n := k; A i 2 = b 1 i 2 b 2 i 2 ; ∀s = 2, ..., n -1, A is,i s+1 = b 1 + ... + b si s i s+1i s b s+1 + i s i s+1 ≥ 0and ∆ is defined by : for any τ = 2, ..., n -10 ≤ i τ ≤ min{b 1 + ... + b τ -max{b 1 , ..., b τ }, i τ +1 }

reg (M i ) is the Castelnuovo-Mumford regularity of M i . If reg(M i ) < d i , for all i = 1, . . . , s then

(1) M 1 ⊗ . . . ⊗M s is a Cohen-Macaulay S 1 ⊗ . . . ⊗S s -module.

(

Proof. We have that for all i = 1, ..., s, The third claim follows from the second and the fact that for all i = 1, ..., s, reg(M

If one of the modules has dimension 0, then we get the following Corollary of Theorem 2.2.

Theorem 4.4. Let S 1 , . . . , S s be graded polynomial rings on disjoints of set of variables. For all i = 1, . . . , s, let M i be a graded finitely generated S i -Cohen-Macaulay module of dimension d i = dim M i . We assume that M i = ⊕ l∈Z M i,l as S i -module, and there is an index

To end this section we exhibit two large classes of ideals that satisfy the hypothesis of Theorem 4.3.

(I) Let NA be a finite generated normal semigroup homogeneous, then by [St][13.14]


Marcel Morales

Dung Thai Nguyen

Nguyen Thi

Veronese transform, and Castelnuovo-Mumford regularity of modules

Keywords: Castelnuovo-Mumford regularity, Veronese ring, Segre ring, Hilbert Series

come

Veronese transform, and Castelnuovo-Mumford regularity of modules

Introduction

Veronese rings, Segre embeddings or more generally Segre-Veronese embeddings are very important rings in Algebraic Geometry. It is well known that these rings are arithmetically Cohen-Macaulay, hence their Hilbert-Poincaré series can be written P R (t) = Q R (t) (1-t) dim R , where Q R (t) is a polynomial on t with Q R (1) = 0 having positive integer coefficients, the sequence of the coefficients of Q R (t) is also called the h-vector of R. The degree of Q R (t) is the Castelnuovo-Mumford regularity (c.f. [START_REF] Eisenbud | The geometry of syzygies. A second course in commutative algebra and algebraic geometry[END_REF][Chapter 4]), and the leading term of Q R (t) is the highest graded Betti number of R. By using very original and elementary methods we are able to compute the leading term of Q R (t). Our results allows to compute the Castelnuovo-Mumford regularity of the n Veronese Module of any finitely generated Cohen-Macaulay graded module, and the rings called of Veronese type.

Note that this result can be proved easily by using local cohomology, but our purpose is to give a very elementary proof.

Our main results improves partially [START_REF] Barcanescu | Betti numbers of Segre-Veronese singularities[END_REF] and [START_REF] Cox | Regularity and Segre-Veronese embeddings[END_REF].

Theorem. Let consider the Segre-Veronese ring R b,n , dim R b,n = b 1 + ...

+ b m + 1. Let P R b,n (t) = Q R b,n (1-t)
dim R b,n be the Hilbert-Poincaré series of R b,n , with Q R b,n = h 0 + h 1 t + ... + h r b,n t r b,n , where r b,n is the Castelnuovo-Mumford regularity of R b,n . We set α b,n = dim R b,nr b,n . After a permutation of b 1 , ..., b m , we can assume that, for all i = 1, ..., m,

⌈ b 1 +1 n 1 ⌉ > b i n i . Then α b,n = ⌈ b 1 + 1 n 1 ⌉ ,
and the highest Betti number of R b,n is

β r b,n = h r b,n = n 1 α b,n -1 b 1 ... n m α b,n -1 b m
Theorem. Fix integers d, n ∈ N * , τ ∈ Z. Let (a l ) l∈Z be a sequence of complex numbers, such that a l = 0 for l << 0, set :

f (t) = l∈Z a l t l , f <n,τ > (t) = l∈Z a nl+τ t l . If f (t) = h(t) (1-t) d with h(t) ∈ C[t, t -1 ] then f <n,τ > (t) = h <n> (t) (1-t) d with h <n> (t) ∈ C[t, t -1 ] such that: • deg h <n,τ > (t) ≤ d -⌈ d-deg h(t)+τ n ⌉,
• If all the coefficients of h(t) are positive real numbers then

deg h <n,τ > (t) = d -⌈ d-deg h(t)+τ n ⌉, • If deg h(t) = d then deg h <n> (t) = d.

Preliminaries on toric rings and Hilbert-Poincaré series

Let S = K[x 0 , ..., x s , x -1 0 , ..., x -1 s ] be a Laurent polynomial ring over a field K on a finite set of variables. For any finite set M of monomials in S, let K[M] ⊂ S be the subring of S generated by the set M. It is the toric ring defined by the semigroup generated by M.

In what follows we consider the special case where S = K[x 0 , ..., x s ] is a polynomial ring over the field K and all the monomials in M are of the same degree.

Example 2.1. Let S = K[x 0 , ..., x b ] ⊕ l∈N S l , and M = {x α 0 0 ...x α b b | α 0 + ... + α b = n}. So that R b,n = K[M] = ⊕ l∈N S nl .
This toric ring is known as the n-Veronese embedding of S.

Example 2.2. More generally, let X 1 , ..., X m , m sets of independent disjoint variables, with Card(

X i ) = b i + 1. Let S i = K[X i ] for i = 1, ..., m, S = K[X 1 ∪ X 2 ∪ X m ], and M = {x 1 x 2 ...x m | x i ∈ X i }. So that R b 1 ,...,bm = K[M] = ⊕ l∈N (S 1 ) l ⊗ ... ⊗ (S m ) l .
This toric ring is known as the Segre embedding of the m polynomial rings S 1 , ..., S m .

Example 2.3. Let X 1 , ..., X m , sets of independent disjoint variables such that X i = {x i,0 , ..., x i,b i }, S i = K[X i ] for i = 1, ..., m, and

n 1 , ..., n m ∈ N, Let S = K[X 1 ∪ X 2 ∪ X m ],
and

M = {x α 1 1 ...x αm m || α i |= n i }, where α i = (α i,0 , ..., α i,b i ), x α i i = x α i,0 i,0 ...x α i,b i i,b i , and | α i |= α i,0 + ... + α i,b i The Segre-Veronese embedding: R b,n = K[M] = ⊕ l∈N (S 1 ) n 1 l ⊗ ... ⊗ (S m ) nml , where b = (b 1 , ..., b m ), n = (n 1 , ..., n m ).
Let S = K[x 0 , ..., x s ] be a polynomial ring over the field K, graded by the standard graduation, that is deg x i = 1, for all i. Let R := S/I, where I ⊂ S is a graded ideal, let M = ⊕ l∈Z M l be a finitely generated graded R-module, hence M is also a S-module. the Hilbert-function of M is defined by H M (l) = dim K M l , for all l ∈ Z, and the Hilbert-Poincaré series of M :

P M (t) = l∈Z H M (l)t l .
It is well known that

P M (t) = Q M (t) (1 -t) dim M ,
where Q M (t) is a Laurent polynomial on t, t -1 with Q M (1) = 0. Moreover if M is a Cohen-Macaulay S-module, all the coefficients of Q M (t) are natural integers,and the Castelnuovo-Mumford regularity of M is the degree of Q M (t). For more details on Hilbert-Poincare series see [START_REF] Morales | Fonctions de Hilbert, genre géométrique d'une singularité quasi-homogène Cohen-Macaulay[END_REF], [4] Remark 2.6. We recall that binomial coefficients can be defined in a more general setting than natural numbers, indeed for k ∈ N, binomial coefficients are polynomial functions in the variable n. More precisely:

(1)

If k = 0 then let n 0 = 1, for all n ∈ C. (2) If k > 0 then let n k = n(n-1)...(n-k+1) k! , for all n ∈ C. Note that for all n ∈ C, n k = (-1) k k-n-1 k and if n ∈ N, n < k, then n k = 0. Example 2.7. Let S = K[x 0 , ..., x b ], be a polynomial ring. Then H S (l) = l+b b if l ≥ 0 0 if l < 0 , P S (t) = 1 (1 -t) b+1 . Note that in fact ∀l ≥ -b, H S (l) = l+b b and 0 = H S (-b -1) = -b-1+b b = (-1) b , so the postulation number of S is -(b + 1). Example 2.8. Let S = K[x 0 , ..., x b ], M = {x α 0 0 ...x α b b | α 0 +...+α b = n}, and R b,n = K[M] the n-Veronese embedding. Then H R b,n (l) = H S (nl) = nl+b b if l ≥ 0 0 if l < 0 . Note that nl+b b = (nl+1)(nl+2)...(nl+b) b! is a polynomial on l with lead- ing term n b l b b! , so that deg(R b,n ) = n b , dim R b,n = b + 1. Note also that ∀l > -⌈ b+1 n ⌉, H R b,n (l) = nl+b b and 0 = H R b,n (-⌈ b+1 n ⌉) = -⌈ b+1 n ⌉n+b b = (-1) b ⌈ b+1 n ⌉n-1 b , so the postulation number of R b,n is -⌈ b+1 n ⌉. More generally the postulation number of R b,n [τ ] is ⌈ b+1+τ n ⌉.

Veronese of generating series

In a recent paper [START_REF] Brenti | The Veronese construction for formal power series and graded algebras[END_REF], Brenti and Walker prove that taking the n-Veronese transform of the h polynomial is a linear function, in this section we improve this result giving a elementary proof of the fact that taking the shifted n-Veronese transform of the h polynomial is a linear function on h. Let recall the following fact:

Theorem 3.1. Let (a l ) l∈Z be a sequence of complex numbers, such that a l = 0 for l << 0, set : f (t) = l∈Z a l t l , TFAE:

• There exists h(t) ∈ C[t, t -1 ] and a natural integer d such that f (t) = h(t) (1-t) d . • There exists Φ(t) ∈ C[t, t -1 ] of degree d -1 with leading coeffi- cient e 0 /(d -1)!, such that Φ(l) = a l for l large enough. Moreover h(1) = e 0 . Definition 3.2. Fix integers d, n ∈ N * , τ ∈ Z.
Let (a l ) l∈Z be a sequence of complex numbers, such that a l = 0 for l << 0, set :

f (t) = l∈Z a l t l , f <n,τ > (t) = l∈Z a nl+τ t l .
By the Theorem 3.

1 if f (t) = h(t) (1-t) d with h(t) ∈ C[t, t -1 ] then f <n,τ > (t) = h <n,τ > (t) (1-t) d with h <n,τ > (t) ∈ C[t, t -1 ].
Let us introduce some notations. To any non zero polynomial

h(t) = h σ t σ + .... + h 0 + h 1 t + ... + h s t s ∈ C[t, t -1 ]
we associate the hvector h = (..., 0, h σ , ..., h s , 0, ...), and we set deg

- → h = deg h(t)
. For j ∈ Z, let -→ ε j be the h-vector of the polynomial t j . Let denote by [t k ]h(t) the coefficient of t k in the polynomial h(t). For any i, j ∈ Z define D i,j by

D i,j = [t in-j ]( (1 -t n ) d (1 -t) d ) = [t in-j ]((1 + t + ...t n-1 ) d ).
Note that

D i,j = Card{(x 1 , ..., x d ) ∈ N d | ∀l, x l ≤ n -1; x 1 + ... + x d = in -j}.
Finally let D[σ, τ ] be the infinite square matrix D[σ, τ ] = (D i+σ,j+τ ). For σ = τ = 0 we write D instead D[0, 0]. We can give some properties of the numbers D i,j .

Lemma 3.3. Let i, j, k ∈ Z, we have:

• D i,j = 0 if either in -j < 0 or in -j > d(n -1).
• For any i, j, D i,j = D d-i,d-j . That is D is symmetrical around the point (d/2, d/2).

• For 0 ≤ k ≤ n -1, D d,d+k = k+d-1 d-1 . • D 1,0 = n+d-1 d-1 -d, and for 1 ≤ k ≤ n, D 1,k = n-k+d-1 d-1 . • For any integers q, k, D d+q,nq+k = D d,k . • For any i, let d -i = nq -k with q = ⌈ d-i n ⌉, 0 ≤ k < n, then D d-⌈ d-i n ⌉,i = k + d -1 d -1 = n⌈ d-i n ⌉ + i -1 d -1 .
Proof. The first claim is trivial. In order to prove the other claims, let remark that the map (x 1 , ..., x d ) → (y 1 , ..., y d ), where y l = (n -1)x l for l = 1, ..., d, establishes a bijection between The fifth claim follows from the equality:

{(x 1 , ..., x d ) ∈ N d | x l ≤ n -1 for l =
(d + q)n -(nq + k) = dn -k.
Finally the sixth claim follows from the third claim, since, if

d -i = nq -k with 0 ≤ k < n, then (d -q)n -i = dn -(d + k), hence D d-q,i = D d,d+k , and n⌈ d-i n ⌉ + i -1 = k + d -1. Remark 3.4. With the notations introduced in 3.2, it is clear that f <n,kn+τ > (t) = t -k f <n,τ > (t), which implies h <n,kn+τ > (t) = t -k h <n,τ > (t)
for any integer numbers k, τ .

The following Theorem improves and gives a simpler proof of [2, Theorem 1.1]: Theorem 3.5. Fix integers d, n ∈ N * , τ ∈ Z. Let (a l ) l∈Z be a sequence of complex numbers, such that a l = 0 for l << 0, set :

f (t) = l∈Z a l t l = h(t) (1 -t) d , f <n,τ > (t) = l∈Z a nl+τ t l = h <n,τ > (t) (1 -t) d ,
where

h(t), h <n,τ > (t) ∈ C[t, t -1 ]. Then for any τ ∈ Z, ------→ h <n,kn+τ > = D[-k, -τ ] h.
Proof. Because the Remark 3.4 we have to compute h <n,τ > (t) only for 0 ≤ τ ≤ n -1. The following formula is clear:

f <n,0> (t n ) + tf <n,1> (t n ) + ... + t n-1 f <n,n-1> (t n ) = f (t), hence h <n,0> (t n ) + th <n,1> (t n ) + ... + t n-1 h <n,n-1> (t n ) (1 -t n ) d = h(t) (1 -t) d , and
h <n,0> (t n ) + th <n,1> (t n ) + ... + t n-1 h <n,n-1> (t n ) = h(t) (1 -t n ) d (1 -t) d , t τ h <n,τ > (t n ) equals the sum of all the terms A β t β of h(t) (1-t n ) d (1-t) d with β ≡ τ mod n. In particular h <n,τ > (t)

is a linear function of h(t).

So it is enough to compute h <n,τ > (t) for the canonical basis {ε j := t j , j ∈ Z} of C[t, t -1 ]. We have

[t i ](h <n,τ > (t)) = [t ni+τ ](h(t) (1 -t n ) d (1 -t) d ), hence ∀j ∈ Z; [t i ](ε <n,τ > j (t)) = [t ni+τ ](t j ) (1 -t n ) d (1 -t) d = [t ni+τ -j ]( (1 -t n ) d (1 -t) d )
which proves our statement.

Corollary 3.6. Fix an integer d ∈ N * . For j ∈ Z, let -→ ε j be the h-vector of the polynomial t j . Then for any n ∈ N * , we have

deg --→ ε <n> j = d -⌈ d -j n ⌉. Moreover the leading coefficient of --→ ε <n> j is n⌈ d-j n ⌉+j-1 d-1
.

Proof. Let remark that the set of t j , j ∈ Z is the canonical basis of C[t, t Let (a l ) l∈Z be a sequence of complex numbers, such that a l = 0 for l << 0, set :

f (t) = l∈Z a l t l , f <n,τ > (t) = l∈Z a nl+τ t l . If f (t) = h(t) (1-t) d with h(t) ∈ C[t, t -1 ] then f <n,τ > (t) = h <n> (t) (1-t) d with h <n> (t) ∈ C[t, t -1 ] such that: • deg h <n,τ > (t) ≤ d -⌈ d-deg h(t)+τ n ⌉,
• If all the coefficients of h(t) are positive real numbers then

deg h <n,τ > (t) = d -⌈ d-deg h(t)+τ n ⌉, • If deg h(t) = d then deg h <n> (t) = d. Proof. Let f (t) = l∈Z a l t l = h(t) (1-t) d , where h(t) ∈ C[t, t -1 ] h(t) = γ σ t σ + ... + γ s t s with deg h(t) = s, γ s = 0. It follows that - → h = s l=σ γ l - → ε l .
We multiply this relation on the left by D[-τ ], so

Theorem 3.5 implies ----→ h <n,τ > = s l=σ γ l --→ ε <n> l-τ . Since deg --→ ε <n> σ-τ ≤ deg ----→ ε <n> σ-τ +1 ≤ ... ≤ deg --→ ε <n> s-τ , we have, deg ----→ h <n,τ > ≤ deg --→ ε <n> s-τ . It is clear that if all the coefficients of h(t)
M <n> = ⊕ l∈Z M nl . Let Q(t) (1 -t) d be the Hilbert-Poincaré series of M , where Q(t) = γ σ t σ + ... + γ s t s ∈ C[t, t -1 ] is the h-polynomial of M , with reg(M ) = deg Q(t) = s. Then • reg M <n> = d -⌈ d-reg M n ⌉.
Moreover by taking the sum over all index l such that ⌈ d-l n ⌉ = ⌈ d-reg M n ⌉, we will get the leading coefficient of Q <n> (t):

l | ⌈ d-l n ⌉=⌈ d-reg M n ⌉ γ l n⌈ d-l n ⌉ + l -1 d -1 . • If reg M ≤ d -1 and n ≥ d then reg M <n> = d -1, and the leading coefficient of Q <n> (t) is d-1 l=0 γ l n⌈ d-l n ⌉ + l -1 d -1 . • If n > reg M ≥ d then reg M <n> = d, and the leading coeffi- cient of Q <n> (t) is reg M l=d γ l l -1 d -1 .
Proof. We have

- → Q = s l=σ γ l - → ε l .
We multiply this relation on the left by D, so Theorem 3.5 implies that for any

n ∈ N * , ---→ Q <n> = s l=σ γ l --→ ε <n> l
. Since γ l ≥ 0 for all l, γ s > 0, and deg

--→ ε <n> σ ≤ deg --→ ε <n> σ+1 ≤ ... ≤ deg --→ ε <n> s , we have, deg ---→ Q <n> = deg --→ ε <n> s = d - ⌈ d-reg M n ⌉, this number is reg(M <n> ) since M <n> is a Cohen-Macaulay S-module. The computation of the leading coefficient of Q <n> (t) is immediate from Lemma 3.6.
4 h-vector of the Segre-Veronese embedding.

The next Theorem improves partially [START_REF] Barcanescu | Betti numbers of Segre-Veronese singularities[END_REF] and [START_REF] Cox | Regularity and Segre-Veronese embeddings[END_REF].

Theorem 4.1. Let consider the Segre-Veronese ring R b,n , dim R b,n = b 1 + ... + b m + 1. Let P R b,n (t) = Q R b,n (t) (1 -t) dim R b,n be the Hilbert-Poincaré series of R b,n , with Q R b,n (t) = h 0 + h 1 t + ... + h r b,n t r b,n , where r b,n = deg Q R b,n (t) is the Castelnuovo-Mumford regularity of R b,n . We set α b,n = dim R b,n -r b,n . After a permutation of b 1 , ..., b m , we can assume that ⌈ b 1 +1 n 1 ⌉ > b i n i ∀i, then α b,n = ⌈ b 1 + 1 n 1 ⌉ r b,n = (b 1 + ... + b m + 1) -⌈ b 1 + 1 n 1 ⌉,
and the highest Betti number of R b,n is

β r b,n = h r b,n = n 1 α b,n -1 b 1 ... n m α b,n -1 b m
Proof. The proof is by double induction on m and b m . The case m = 1 is given by the Example 2.8 and Corollary 3.6, so we can assume m ≥ 2. We have that

⌈ b 1 +1 n 1 ⌉ > bm nm > bm-
of R b,n is H R b,n (l) = n 1 l+b 1 b 1 ... nml+bm bm , so H R b,n (l) = (1 + n m b m l)H R b-ǫm,n (l). ( 1
)
Let P R b-ǫm,n (t) = Q R b-ǫm,n (t) (1 -t) b 1 +...+bm be the Hilbert-Poincaré series of R b-ǫm,n , where Q R b-ǫm,n (t) = h 0 + h 1 t + ... + h r b-ǫm,n t r b-ǫm,n
, with h r b-ǫm,n = 0. In order to avoid any confusion we also set :

P R b,n (t) = Q R b,n (t) (1 -t) b 1 +...+bm+1 be the Hilbert-Poincaré series of R b,n , where Q R b,n (t) = ĥ0 + ... + ĥr b,n t r b,n ,
with ĥr b,n = 0.

Let β = nm bm , by simple calculations from (1) we get:

P R b,n (t) = P R b-ǫm,n (t) + βtP ′ R b-ǫm,n (t). (2)
Hence dim R b,n = dim R b-ǫm,n + 1, and

Q R b,n (t) = Q R b-ǫm,n (t)+t[Q R b-ǫm,n (t)(βR b-ǫm,n -1)+βQ ′ R b-ǫm,n (t)-βtQ ′ R b-ǫm,n (t)], note that Q R b,n (1) = β dim R b-ǫm,n Q R b-ǫm,n (1) = 0.
In particular we have r b,n ≤ r b-ǫm,n +1 and for all k = 0, ..., r b-ǫm,n + 1 we have ĥk

= h k-1 ( n m b m dim R b-ǫm,n -(k -1) n m b m -1) + h k (k n m b m + 1). ( 3
)
By induction hypothesis we have α b-ǫm,n = ⌈ b 1 +1 n 1 ⌉ = bm nm , so we put k = r b-ǫm,n + 1 in equality (3), and we get:

ĥr b-ǫm,n +1 = h r b-ǫm,n ( n m α b-ǫm,n -b m b m ) = 0.
Hence ĥr b-ǫm,n +1 is the leading coefficient of Q R b,n and r b,n = r b-ǫm,n + 1 and

α b,n = α b-ǫm,n = ⌈ b 1 +1 n 1 ⌉. By induction hypothesis h r b-ǫm,n = n 1 α b,n -1 b 1 ... n m-1 α b,n -1 b m-1 n m α b,n -1 b m so that ĥr b,n = n 1 α b,n -1 b 1 ... n m-1 α b,n -1 b m-1 n m α b,n -1 b m -1 ( n m α b,n -b m b m ) = n 1 α b,n -1 b 1 ... n m-1 α b,n -1 b m-1 n m α b,n -1 b m .

Rings of Veronese type

Let b, n ∈ N * , a = (a 0 , ..., a b ) ∈ N b+1 such that 1 ≤ a i ≤ n, a 0 + ... + a b > n, and M b,n,a be the set of monomials of the polynomial ring K[x 0 , ..., x b ]:

M b,n,a = {x α 0 0 ...x α b b | α 0 + ... + α b = n, α i ≤ a i , ∀i = 0, ..., b}.
Let denote by R b,n,a the toric subring of K[x 0 , ..., x b ] generated by M b,n,a . It is well known that R b,n,a is a Cohen-Macaulay ring. Let S be the collection of subsets of {0, ..., b} such that : S ∈ S if and only if S ⊂ {0, ..., b}, and ΣS := i∈S a i < n. Our aim is to study the Hilbert-Poincaré series of R b,n,a :

P R b,n,a = S∈S (-1) |S| l≥0 l(n -ΣS)-| S | +b b t l .
The following Corollary follows immediately from 3.6.

Corollary 5.2. For any S ∈ S and k ∈ N * , we have:

l≥0 kl-| S | +b b t l = Q S,k (t) (1 -t) b+1 , where Q S,k (t) is a polynomial with Q S,k (1) = 0, with leading term kα S,k + | S | -1 b t b+1-α S,k , with α S,k = ⌈ b+1-|S| k ⌉.
The following theorem is immediate from 5.1 and Corollary 5.2. It improves the description of the Hilbert Poincaré series given in [START_REF] Katzman | The Hilbert series of algebras of the Veronese type[END_REF]. Theorem 5.3. With the above notations, let S be the collection of subsets of {0, ..., b} such that : S ∈ S if and only if S ⊂ {0, ..., b}, and ΣS := i∈S a i < n. Then we can write the Hilbert-Poincaré series of R b,n,a : In general leading terms of the alternating sum can cancel, as we can see in the next example.

P R b,n,a = Q b,n,a (t) (1 -t) b+1 , with Q b,n,a (t) = S∈S (-1) |S| Q S,n (t), where Q S,n (t) is a polynomial with Q S,n (1)

Remark 2 . 5 .

 25 [Chapter 4], [7][Chapter 4]. Theorem 2.4. (Hilbert's Theorem) let M = ⊕ l∈Z M l be a finitely generated graded S-module. There exists a polynomial with integer coefficients Φ H M (l) such that H M (l) = Φ H M (l), for l large enough. Moreover the leading term of Φ H M (l) can be written as : deg(M ) d! l d , where d + 1 is the dimension of M and deg(M ) the degree or multiplicity of M . The postulation number of the Hilbert function is the biggest integer l such that H M (l) = Φ H M (l). It is well known , ([10], [4][Chapter 4]), that the postulation number equals the degree of the rational fraction defining the Poincaré series.

1 ,

 1 ..., d; x 1 + ... + x d = in -j} and {(y 1 , ..., y d ) ∈ N d | y l ≤ n-1 for l = 1, ..., d; y 1 +...+y d = (d-i)n-(d-j)}. The third claim follows from the second claim, because if 0 ≤ k ≤ n -1, then the sets {(x 1 , ..., x d ) ∈ N d | x l ≤ n-1 for l = 1, ..., d; x 1 +...+x d = dn-d-k} and {(y 1 , ..., y d ) ∈ N d | y 1 + ... + y d = k} are in bijection. The fourth claim follows trivially from the previous items.

1 . 1 . 3 . 7 . 1 Theorem 3 . 8 .

 1137138 -1 ]. We have byTheorem 3.5 that D -→ ε j = j column vector of D. By the Example 2.8, we have that deg The last claim follows from the Lemma 3.3. Indeed for any j ∈ Z, we have D d-⌈ d-j n ⌉,j = n⌈ d-j n ⌉+j-1 d-This Example Let d = 2 and n ∈ N * , we can describe the matrix D ij -(n+1) ... Fix integers d, n ∈ N * , τ ∈ Z.

Theorem 3 . 9 .

 39 are positive real numbers then deg ----→ h <n,τ > = deg --→ ε <n> s-τ . In the special case s = d, we have seen that for 0 ≤ l ≤ d -1 and any n ∈ N * , deg --→ ε <n> l = d -⌈ d-l n ⌉ ≤ d -1, and deg Let n ∈ N * , S be a standard graded polynomial ring, M = ⊕ l∈Z M l be a finitely generated Cohen-Macaulay graded S-module of dimension d ≥ 1, and

  1 nm , so by induction hypothesis the theorem is true for R b-ǫm,n , where bǫ m = (b 1 , ..., b m-1 , b m -1). On the other hand the Hilbert function

Theorem 5 . 1 .

 51 ([9]) With the above notations the Hilbert-function of R b,n,a is ∀l ≥ 0; H b,n,a (l) = S∈S (-1) |S| l(n -ΣS)-| S | +b b We have dim(R b,n,a ) = b + 1, and its degree or multiplicity is deg(R b,n,a ) = S∈S (-1) |S| (n -ΣS) b .

(- 1 )= 0 2 . 2 .

 122 = 0, with leading term (n -ΣS)α S,n-ΣS + | S | -1 b t b+1-α S,n-ΣS , where α S,n-ΣS = ⌈ b+1-|S| n-ΣS ⌉. Part one of the following corollary improves [9][Cor. 2.12]. Corollary 5.4. With the above notations: 1. reg(R b,n,a ) ≤ b + 1 -⌈ b+1 n ⌉, and the equality is true if and only ifS∈S,α S,n-ΣS =⌈ b+1 n ⌉ |S| (n -ΣS)α S,n-ΣS + | S | -1 b If b + 1 > n 2 then reg(R b,n,a ) = b + 1 -⌈ b+1 n ⌉. Moreover the leading term of Q b,n,a (t) is (n⌈ b+1 n ⌉-1 b t b+1-⌈ b+1 n ⌉ , Proof. 1. It is enough to prove that min S∈S ⌈ b+1-|S| n-ΣS ⌉ = ⌈ b+1 n ⌉. We consider two cases, • if b + 1 < n then ⌈ b+1 n ⌉ = 1 ≤ ⌈ b+1-|S| n-ΣS ⌉, ∀S ∈ S. • If b + 1 ≥ n, then b + 1 n ≤ b + 1-| S | n -ΣS ⇔ (b+1)(n-ΣS) ≤ n(b+1-| S |) ⇔ (b+1)ΣS ≥ n | S |,this is true since by hypothesis b+1 n ≥ 1 ≥ |S| ΣS . Let b + 1 > n 2 and S = ∅. By definition ⌈ b+1 n ⌉ is the integer q such that b + 1 = qnr, with 0 ≤ r < n and q ≥ n + 1. We have b + 1-| S |= qnr-| S |= q(n -ΣS)r-| S | +qΣS, and qΣS-| S |≥ (n + 1)ΣS-| S |≥ nΣS > r, so that qΣS-| S | -r > 0, hence ⌈ b+1-|S| n-ΣS ⌉ > q = ⌈ b+1 n ⌉.

Example 5 . 5 . 1 ⌉ = 3 .

 5513 Let consider the ring R 4,3,(1,1,1,1,1) , the sets S can have 0,1 or 2 elements, and we have:If S = ∅ then α ∅,3 = ⌈ 5 3 ⌉ = 2, if | S |= 1 then α S,3 = ⌈ 4 2 ⌉ = 2,and finally if | S |= 2 then α S,2 = ⌈ 3 By using Theorem 5.3 we can write P R 4,3,(1,1,1,1,1) = Q 0 (t) -5Q 1 (t) + 10Q

  , with Q 0 (t) = 5t3 + ...; Q 1 (t) = t 3 + ...; Q 2 (t) = t 2 + ... Note that in this case Q 0 (t) -5Q 1 (t) + 10Q 2 (t) = h 0 + h 1 t + h 2 t 2 , where h 0 = 1, h 1 = 5 and since h 0 + h 1 + h 2 = deg(R 4,3,(1,1,1,1,1) ) = 11, we get h 2 = 5, so that P R 4,3,(1,1,1,1,1) = 1 + 5t + 5t 2 (1t) 5 .

2 (t) (1t)

5

,


D J Hykel

Z S Wang

P Castellazzi

T Crozes

G Shaw

K Hasselbach

K Schuster

MicroSQUID Force microscopy in a dilution refrigerator *

We present a new generation of a scanning MicroSQUID microscope operating in an inverted dilution refrigerator. The MicroSQUIDs have a size of 1.21 µm 2 and a magnetic flux sensitivity of 120 µΦ 0 / √ Hz and thus a field sensitivity of 550 µG/ √ Hz. The scan range at low temperatures is about 80 µm and a coarse displacement of 5 mm in x and y direction has been implemented.

The MicroSQUID-to-sample distance is regulated using a tuning fork based force detection. A MicroSQUID-to-sample distance of 420 nm has been obtained. The reliable knowledge of this distance is necessary to obtain a trustworthy estimate of the absolute value of the superconducting penetration depth. An outlook will be given on the ongoing direction of development.

INTRODUCTION

Scanning probe microscopy has been widely used for the study [START_REF] Kirtley | [END_REF] of superconductors, we briefly discuss advantages of each of the most common methods. Magnetic force microscopy compared to other scanning techniques has the huge advantage of its high spatial resolution of ∼ 50 nm, and moreover recently significant progress has been made increasing its magnetic sensitivity [2]. The vortex/tip interaction is not negligible and can lead to displacement of vortices. By controlling the height of the MFM tip the displacement of vortices can be turned on and off and thus it can be used to control the vortex position. In order to interpret the MFM images quantitatively precise information of the tip magnetization is required. Another scanning technique is the Hall probe microscopy [3], [4]. It combines good spatial resolution (850 nm) with a good magnetic resolution of ∼ 2.9 × 10 -3 G/ √ Hz. The Hall probe sensitivity is nearly independent of the Hall cross area (see [5]), but is limited by Johnson noise, 1/f noise and the current density. Hall probe microscopes are very attractive for their operation in wide temperature and field ranges.

SCANNING SQUID MICROSCOPY

As Superconducting Quantum Interference Devices (SQUIDs) are flux sensors their field sensitivity varies linearly with the pick-up loop area. Therefore field sensitivity and spatial resolution can be optimized for the given experiment by choosing the size of the flux pick-up loop.

For scanning SQUID microscopy (SSM) two different approaches are pursued for the design of SQUID sensors. The first approach consists in scaling down the size of highly sensitive conventional niobium tunnel junctions SQUIDs. The flux noise of these SQUIDs is of the order of 0.7 µΦ 0 / √ Hz which can be translated into a field noise of the order of 15 µG/ √ Hz. at 4.2 K. Operated without z-feedback loop, the SQUID chip is in contact with the sample surface [6], making scanning at close distances difficult. The fabrication of multilayer Nb SQUIDs is very demanding and their wide Nb structures are prone to vortex pinning in magnetic fields of a few gauss, thus limiting the field-range of operation of these SQUIDs. If the SQUID is efficiently cooled, it is possible to image vortices in samples at temperatures of 100 K and more, as long as the sample is only in weak thermal contact with the SQUID [7]. The other approach consists in choosing a simpler SQUID design, aimed at increased spatial resolution. The most recent development is the SQUID on tip (SOT) design [8], [9]. Here a SQUID is deposited on the edge of a pulled quartz tube. SQUIDs with an effective diameter of less than 100 nm can be fabricated by three successive operations of evaporation. The quartz tube is scanned perpendicularly to the surface, in consequence the SQUID leads are parallel to the applied field, and the SOT can operate in fields up to 1 T [9].

The SQUID readout relies on the use of a low temperature SQUID amplifier array conferring the detector a very high flux sensitivity of 50 nΦ 0 / √ Hz which can be approximated by a field sensitivity of ∼ 50 µG/ √ Hz [9] for a 160 nm diameter lead SOT. The SQUID-sample distance can be controlled by attaching [10] the metalized quartz tube to a tuning fork.

SCANNING MICROSQUID MICROSCOPE

In the following we describe our approach of fabricating and using planar MicroSQUIDs, taking advantage of the power of lithography and achieving routinely high spatial resolution.

An array of hundreds of MicroSQUIDs is patterned by e-beam lithography on a silicon wafer. The Josephson Junctions are realized as Dayem-bridges of typically 20-30 nm width.

The MicroSQUIDs are routinely made of aluminum by a lift-off process. After this step a photoresist mask defining the contour of each MicroSQUID chip is precisely aligned with the MicroSQUID array. This mask protects the MicroSQUIDs during the Si deep etch, which removes most of the Si in proximity of the MicroSQUID [11]. The precise etch simplifies the positioning of the MicroSQUID on the quartz tuning fork used for topographical imaging and height control [12]. This geometry has the advantage of being mechanically robust, as the MicroSQUID-to-tip distance protects the MicroSQUID from the wear and tear of scanning, allowing us to scan in the topographic feedback mode for several weeks.

We use custom made room temperature electronics generating current ramps for the detection of the critical current of the MicroSQUID. As soon as the bias-current ramp reaches the critical current of the device, the bias-current is set back to zero in less than 25 nanoseconds. Once in the normal state an energy of the order of 4 × 10 -15 joule is deposited in the MicroSQUID (R N =100 Ω, I c of 40 µA). The deposited energy is easily removed by the connecting leads and the measurements can be repeated up to 10000 times per second. The present MicroSQUID has a flux noise of 120 µΦ 0 / √ Hz and can detect the stray field with a field noise of ∼ 550 µG/ √ Hz. In the following we will present the new implementation of our MicroSQUID microscope and show images validating the approach.

Our new microscope [13], [14] is placed in an inverted dilution refrigerator [START_REF] Wernsdorfer | Inverted Dilution Refrigerator[END_REF] The microscope has three piezo-electrically driven linear stages for coarse displacements along the xyz-directions of the order of 5 mm (Fig. 1). The z-direction stage encloses a hollow titanium prism containing a xy-scanner [START_REF] Siegel | [END_REF] allowing for a scan range of 80 µm in the x-and y-directions at 0.2 K, the base operating temperature. The piezo-electrical tuning fork is mechanically excited with a "thickness" piezoceramic. At room temperature the tuning fork current is amplified with a custom-built current/voltage converter of gain 10 8 V/A, and the voltage is demodulated with a fast lockin-amplifier [START_REF]Signal Recovery Ametek: Model 7280[END_REF]. The phase signal of the lockinamplifier enters a phase locked loop, changing the excitation frequency in order to maintain the tuning fork at resonance frequency. The difference of the actual resonance frequency from the set frequency enters a PID regulation loop regulating the distance between the SQUID tip and the sample surface. The distance is controlled by a z-piezo stack [START_REF][END_REF] (C ∼ 1 µF) attached on the xy-stage [19] and the regulation is implemented on a digital signal processing card [20]. The bandwidth of the regulation is about 1000 Hz, which must be degraded to 100 Hz at the lowest temperature in order to limit heating due to the charging of the z-piezo stack. The usual scan speed is 80 µm in 14 seconds. During a scan of 256 pixels 7680 critical current measurements are accumulated at 830 Hz which takes already 9 seconds out of the 14 seconds. The MicroSQUID is always thermally anchored to the mixing chamber, the sample stage on the scanner is only weakly thermally anchored to the mixing chamber or the still, allowing for fast thermal cycling of the sample above its superconducting transition temperature.

Scanning magnetic probe microscopes can measure the magnetic stray field of a single vortex, and are thus employed to derive absolute values of the superconducting penetration depth. The absolute value of the penetration depth is of importance as it also enters in the expression of the penetration depth temperature dependence, which is used as a probe for the symmetry of the superconducting order parameter. The z component of the magnetic stray field of a vortex, h z , measured in a plane at a height z p , depends in the same manner [21] on this height and the penetration depth, λ:

∂h z ∂λ = -2 ∂h z ∂z p (1)
The reliable knowledge of the height z p is a prerequisite for deriving the superconducting penetration depth from the fitting of the field profile of a vortex. In our case z p is the MicroSQUID-to-sample distance. The aim is to gain a precise control of this distance. A MicroSQUID-to-sample distance of 420 ± 30 nm can be achieved under the condition that the MicroSQUID is placed at about 5 µm away from the touch down point (white line on the left panel of Fig. 2 and that the angles of approach are small and well controlled. The other parameter entering the penetration depth determination is the displacement calibration of the xy-piezo-scanner. For this purpose we have designed a self-similar chessboard pattern of Nb squares of 100 µm, 10 µm, and 1 µm a side, alternating with empty squares of the same sizes. The pattern has been obtained by reactive ion etching of a 200 nm thick Nb film, giving also rise to topographic contrast. The pattern contains orientational marks (angles) indicating the direction to the center of the pattern (see Fig. 3).

The topographic image Fig. 3(e) of the calibration pattern is convolved with the shape of the probe-tip, (Fig. 2, left), resulting in a softening of the edges of the 10 µm squares (dark grey) in the topographic image. As the scan angle is shallow, the tip does not touch the bottom of the pattern between the squares. The tip does touch the bottom of the pattern further away from the squares (clear grey). The magnetic image acquired simultaneously (Fig. 3(f)) appears slightly shifted in upwards direction. The scan direction is from the top toward the bottom, and the tip is oriented towards the top of the image (Fig. 3(h)), consequently the MicroSQUID senses first the magnetic contrast at a given point before the tip senses the topographic signal at this point, resulting in the observed shift in the images.

The magnetic image is an image of the critical current of the MicroSQUID as function of position. In the magnetic image the ambient magnetic field gives rise to gray contrast, magnetic shielding at the location of the Nb squares appears dark and vortices, locations of high amplitudes of the stray field, appear bright. In looking closely at the close-up view of the 10 µm square, (Fig. 3(g)), distinct 1 µm squares appear at the edge, surrounded by a bright halo, due to the compression of magnetic flux by the superconducting Nb. From these calibration images we deduce an image size of 85 ± 1 µm for the vertical direction and 70 ± 1µm for the horizontal direction.

DISCUSSION AND CONCLUSION

MicroSQUIDs operation close to the quantum limit has been demonstrated [6], [9], [22].

In order to obtain a measuring device the question of how to couple the magnetic signal of the sample into the SQUID becomes important. In static SQUID experiments it is possible to place magnetic nanoparticles as close as possible to nanometer wide constrictions in the superconducting loop, maximizing the flux coupling. In such a geometry the actual SQUID diameter is less relevant than the width of the constriction [23], [24]. The challenge of microscopy consists in scanning the probe above the surface and thus a minimal MicroSQUID-to-sample distance must be maintained. Moreover the smaller the pickup loop is, the closer the probe has to be in order to couple a significant amount of the flux source into the loop. The signal diminishes rapidly as function of MicroSQUID-to-sample distance for dipole, monopole and line sources of magnetic fields [START_REF] Kirtley | [END_REF]. A further difficulty arises from the fact that the microscope has to work in an unshielded environment at cryogenic temperatures.

The microscope presented here has a tuning fork based height allowing us to scan several samples and calibration patterns during one cool-down lasting several month. Imaging at the base temperature of 0.2 K was achieved for the study of the ferromagnetic superconductor UCoGe [25] . We showed that by controlling the shape of the very end of the MicroSQUID-tip we can reduce the MicroSQIUID-sample distance to at least 420 nm while scanning reliably, opening the path to the precise determination of the penetration depth in superconductors.

MicroSQUIDs have a huge potential for increase in sensitivity. We have recently reported on the continuous readout of hysteretic Al/Nb/W SQUIDs [26] in an magnetically unshielded environment, reaching a noise level of 5 × 10

Figure 1 .

 1 Figure 1. Photos of the microscope. Left: the completely assembled microscope, center: close-up of the tuning fork with the SQUID-chip underneath, right: the top part of the microscope with the sensors (top) and the lower part with the sample stage (bottom).

Figure 2 .

 2 Figure 2. Left:. SEM Image of a deep etched Si tip carrying a highlighted MicroSQUID having an inner diameter of 1µm, the touch down point (white line) is selected in tilting the SQUID by 2 sidewise (center) and 4 downward (right), thus a MicroSQUID-to-sample distance of 420 ± 30 nm can be obtained.

Figure 3 .

 3 Figure 3. Sample: (a,b) optical images of the calibration pattern consisting of niobium squares of lateral length 100 µm, 10 µm and 1 µm. (c,d) SEM images: zoom on one small square of (b), (d) zoom on the smallest Nb square (1µm), (e) is the topographic image obtained by recording the tuning fork signal, (f) is the simultaneously acquired magnetic image (T = 0.45 K) , (g) is the close-up view of a 10 µm square with one vortex in a 1 µm square, (h) schematized top view of the orientation of the probe-tip during scanning over the sample, represented by the black square, (not to scale).

  -5 Φ 0 / √ Hz or ∼ 130 µG/ √ Hz for a square MicroSQUID of 2.5 µm side length. This continuous readout will allow us to implement an on-chip feedback loop and to use highly sensitive SQUID electronics for the MicroSQUIDreadout. We are currently developing a low temperature sample displacement (T < 0.1 K) mechanism, driven by room temperature motors, adapted for a novel inverted dilution refrigerator, built and tested recently. This design will remove most of the piezoelectric high voltage lines from the SQUID environment, reducing possible sources of noise and leveling the path to user friendly low temperature scanning MicroSQUID microscopy. * P. C. and K. H. acknowledge support in the framework of MICROKELVIN, the EU FRP7 low-temperature infrastructure under Grant No. 228464. Z.S. W. acknowledges thankfully the support of the French Embassy in Beijing and the CROUS. This work has been supported by the French National Research Agency, Grant No. ANR-09-Blanc-0211 SupraTetrafer and

  on top of a vibration isolated table. A single room temperature seal is sufficient to close the cryostat.The refrigerator is inverted in the sense that the sample space is on top of the refrigerator, approximately 40 cm above the table-top, a hundred liter helium dewar is suspended under the table supplying liquid He to the 4.2 K pot of the refrigerator. If the mixing chamber stage is unoccupied, cool downs to 4.2 K can be achieved in two hours and two hours more are sufficient to condense 35 liter of mixture, with a base temperature below 40 mK. Once the refrigerator is operating, the liquid helium consumption is less then 10 liters per day.