Annales de l’institut Fourier (AIF)

Rosay, Jean-Pierre

Stout, Edgar Lee

ANNALES DE L’INSTITUT FOURIER
JEAN-PIERRE ROSAY
EDGAR LEE STOUT
An approximation theorem related to good
compact sets in the sense of Martineau
Annales de l’institut Fourier, tome 50, no 2 (2000), p. 677-687
<http://www.numdam.org/item?id=AIF_2000__50_2_677_0>
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Ann. Inst. Fourier, Grenoble
50, 2 (special Cinquantenaire) (2000), 677-687
AN APPROXIMATION THEOREM RELATED TO
GOOD COMPACT SETS
IN THE SENSE OF MARTINEAU
by Jean-Pierre ROSAY<1) & Edgar Lee STOUT
This paper is about a theorem (Theorem 1) stated, but in our opinion
not fully proved, by J.E. Bjork and a theorem (Theorem 2) proved but not
stated by E. Bishop.
In [2] Bjork states the following theorem (in a footnote):
THEOREM 1. — Let U be an open set in C71, and let K be a
compact subset ofU. Then there exists a neighborhood W of K such that
if f C 0(U) and if, for every I € N71, Q^- is the uniform limit on K of a
sequence of polynomials, then f is the uniform limit on W of a sequence
of polynomials.
In this paper 0(U) denotes the algebra of holomorphic functions on
U and polynomial will always mean holomorphic polynomial.
Theorem 1 has an important corollary, called the Main Theorem in
[2], which says that every compact set in C77' is a good compact set in the
sense of Martineau [5]. It implies, by an argument given in [5], that every
compact subset of every Stein manifold is a good compact set.
COROLLARY. — With K and U as above, if f € 0(U), and if f is
uniformly approximable by polynomials on some neighborhood of K, then
f is uniformly approximable by polynomials on the fixed neighborhood W.
^ Research supported in part by a grant from the NSF.
Keywords: Polynomial hulls - Polynomial approximation - Analytic functionals - Good
compact sets.
Math. classification: 32E20 - 32E30 - 46F15 - 46A22.
678 J.-P. ROSAY & E.L. STOUT
This approximation property is a necessary and sufficient condition in
order to extend every analytic functional carried by K to a "local analytic
functional", which is useful for decomposing analytic functionals into sums
of analytic functionals carried by smaller sets. See [5], Theorem 1.2 in
Chapter 1 and 2.A in [6].
Bjork's arguments do not differ substantially from Martineau's argu-
ment, establishing the theorem under an additional hypothesis [5], Theo-
rem 1.1' in Chapter 1. In our view, Bjork's proof has a gap in that it does
not establish a compactness property of the topology that he introduces on
a maximal ideal space. (Alternatively, it has to be shown that, on the poly-
nomial hull, this topology coincides with the C71 topology.) This is however
the heart of the matter. (In similar situations see [1] page 48, lines 19-25,
or [5].)
It is true that Bjork himself gives an indication leading to a complete
proof. On page 495 in [2], he mentions Bishop's work, claiming that "we
shall not need this result". Our point is that, apparently, one does need it.
The next, and not at all easy, question is: Which result of Bishop do
we need? The paper [1] is extremely deep but hard to read. The following
theorem is not to be found stated in [I], nor in [3] or [4], where a very helpful
introduction to part of Bishop's work is given. Nonetheless, we undoubtedly
can attribute the result to Bishop.
THEOREM 2. — Let L be a compact set, and let U be an open set
in C71 with L C U. Let f <E 0(U). Let I/i be the graph in C14-1 defined by
L^{{zJ{z))^Cn^:z^L}.
Let Li denote the polynomially convex hull ofLi. For every e > 0, there
exist k € N and an exceptional set Ee C C71 of measure not more than
e such that, for every z € C"' — Ee, the fiber of L\ over z has at most k
elements. Moreover k depends only on L, U and e, not on /.
The fiber of Li over z is the set of (z,w) € C^1, w G C, such that
{z,w) e Li.
We propose this statement (which could easily be generalized to
Riemann domains) as an extremely readable one, one that may well suffice
for many applications of Bishop's theory, while avoiding extensions of
norms, etc.
In Section 2 we establish Theorem 2, but assuming Theorem 2 we
first prove Theorem 1 in Section 1. Although the proof has been sketched
ANNALES DE L'lNSTITUT FOURIER
GOOD COMPACT SETS 679
in [6] (Appendix on good compact sets) it seems desirable to have a clear
complete proof written down for such an important result.
We should point out the paper [7], by W.R. Zame. In that paper the
author has an interesting systematic discussion of algebras invariant under
differentiation. He uses considerably more of Bishop's theory than we do.
A main result in [7] is Theorem 4.1. It is claimed that a slight modification
of its proof, which, the reader should be warned, is marred by misprints,
leads to Theorem 4.2 in [7], a special case of which is Theorem 1 of this
paper.
1. Proof of Theorem 1.
It is enough to prove that there exists a neighborhood W\ of K^ the
polynomial hull of K in C72, such that for any / as in the statement of the
theorem, there exists / € 0(W\) such that f = f near K. The theorem
follows then from the Oka-Well Theorem.
The proof will be presented in three steps.
In Step 1, we recall some basic facts on algebras of holomorphic
functions closed under differentiation. (See [1] Section 4.) In Step 2,
consequences are drawn from Theorem 2. In Step 3, we apply the above to
the situation of Theorem 1. Some details in the proof are left to the reader
who can also look at [2].
Step 1). For an open subset U of C72, we consider a subalgebra A
of 0(U), that contains the polynomials and such that for every / G A
and for every j e {1 , . . . ,n}, |̂ - 6 A. If H is a compact subset of U,
we denote by SpeCfj A the set of algebra homomorphisms from A onto
C that are continuous with respect to sup norm on H. The coordinate
functions in C^ are denoted Zi , . . . ,Zn. If h € SpeCfj A, then ^(h) is
the point in C71 defined by 7r(h) = (/i(Zi),... ,/i(Z^)). We also write
7T(h)=Z={z^...^n)=h(Z).
Fix compact sets K C L C U, with K included in the interior of L.
Let r = dist(J^, U \ L), a positive number. We shall establish the following
two simple facts i) and ii), which are familiar and basic to these kinds of
considerations.
i) If h € Spec^- A and h(Z) = z, there exists a holomorphic map (/?,
a section of TT, of the ball B{z,r\ centered at z with radius r, into
TOME 50 (2000), FASCICULE 2 (special Cinquantenaire)
680 j.-p. ROSAY & E.L. STOUT
Spec^ A,
y?: B(z,r) -^Spec^A,
such that TT o <^ = id and (p(z) = /i.
That y? is a holomorphic map means that for every / € A, the map
z ̂  ^p(z){f) is a holomorphic function.
This is an immediate consequence of the Cauchy estimates on deriva-
tives.
For ^ C B(z,r), simply set
1.«)(/)'E^(^)«-)-
ii) If (p1 is another such map, associated to a homomorphism h' (possibly
h itself) with h'(Z) = z ' , then: either ^p = ̂  on B(z,r) H B(^',r) or
(p{B(z,r)) and (/?'(B(^,r)) are disjoint.
Indeed if (^(Q = y/«-), then not only ^(C)(/) = ^(C)(/) for all
/ C A, but also |^(^)(/)) = ^(^(y)) for all multi-indices J, so
^(C)(/) = ̂ (0(/) on B(^r) nB(^r).
Step 2). In the setting of 1), Theorem 2 has the following consequence:
COROLLARY. — There exists k e N such that for every z e C71, the
fiber ofSpec^ A over z has at most k elements.
Proof of the corollary. — Although we stated the corollary as it is
to be used, for Spec^ A, we will rather prove it for Spec^ A, just in order
to keep the same notations. This is of course inconsequential.
Let r > 0 be as in 1), let |B(0,r)| denote the volume of the ball
B(0,r), let e < |B(0,r)|, and fix k as in Theorem 2. (The choice of k
depends, in part, on that of e.) If the fiber of Spec^ over z had {k + 1)
distinct points / i i , . . . , hk+i, there would exist / € A such that
^•(/)^M/) for l < ^ j < £ ^ k + l .
For each j e {1 , . . . . k + 1} let then ̂  be a holomorphic map defined on
B(z,r) as in 1) with ^pj(z) = hj.
Let
Li = {(CJ(C))C C^1 : < € L}.
For every C € B(z,r), the restriction of ^(C) to the algebra generated by
Z i , . . . , Zn and / defines an element ^(C) € Li, which is in the fiber of Li
ANNALES DE L'lNSTITUT FOURIER
GOOD COMPACT SETS 681
over C. (Indeed the algebra of functions on L generated by Z\,..., Zn and
/ can be identified with the algebra of functions on L\ generated by the
polynomials, whose maximal ideal space is Li.) Except on a set of measure
not more than e, this fiber over C, has at most k elements.
Since e < |B(z,r)|, there exist l ^ j < ^ ^ A ; + l , and a set G of
positive measure in B(z,r) such that for every C ^ G:
^(OOO-^CX/)-
It follows that ^j(C)(/) = ^(C)(/) for au C ^ B(z,r), in particular for
^ == 2;, a contradiction.
Step 3). Following [2], we apply 1) and 2) to the algebra A of
holomorphic functions on U that can be uniformly approximated on K
by polynomials and whose derivatives can also be so approximated.
If z C K (the polynomial hull of K in C71) the map P ̂  P{z} defined
on the algebra of polynomials, extends naturally to a homomorphism of A
onto C, that we denote by hz. This is the natural inclusion K C Spec^- A.
Fix L as in 1), with K C intL (c U. For each z 6 K, denote by ^z
the map from B(z^r) into Spec^ A that satisfies ^pz{z) = ̂  as defined in
1). If ^ € K, then for C e B(^, r) we have <^(C)(/) = /(C) for all / in our
algebra.
The following lemma corresponds to what we consider to be the gap
in [2]:
LEMMA 1. — There exists 6 > 0 such that if z and z ' E K and
\z — z'\ < 6 then hz' = ^(^/).
Proof of the lemma. — Using the uniqueness result in ii (Step 1)
and the Bolzano-Weierstrass Theorem, it is enough to show that if {zj) is
a sequence of points zj € K with limit z € K^ then for j large enough
hz =^,(^).
Indeed by Schwarz's Lemma, ( p z - ( ^ ) —^ hz. However (pz . ( z ) belongs
J-^oo J
to the fiber of Spec^ A over z^ which is a finite set, so the sequence must
be stationary.
This argument is taken from [1] page 488, lines 19-25.
End of the proof of Theorem 1. — The proof ends exactly as in [I],
[2], [5]. For every z € K, and every / € A the function C \—> (pz (€)(/) ls a
TOME 50 (2000), FASCICULE 2 (special Cinquantenaire)
682 J.-P. ROSAY & E.L. STOUT
holomorphic function denned for |C — z\ < r, such that </^(C)(/) = /(C) lr
z e K. With 6 as above, set 6\ == min(r, j). Denote by ̂  the restriction
of (/^ to the ball B{z,6\). It follows from the lemma above and from the
uniqueness result 1 ii) that the functions i^z agree: On the intersection
of their domains ^z =- ^ z ' - So the functions C '—^ ^z (€)(/) define a
holomorphic function on the 6\ neighborhood of^in C71, which agrees with
/ on some neighborhood of K. Since 6^ does not depend on /, Theorem 1
follows from the Oka-Weil Theorem.
Remarks. — 1) Theorem 2 has allowed us to overcome a problem
of multivaluedness, which is the main difficulty in the proof. The proof
is otherwise rather immediate. The problem of multivaluedness was the
difficulty met by Martineau.
2) For 0 < e < 1 consider:
K = {(e10, 0) € C2} U {(4e^, e) e C2}
and
U={(z^z^eC2:\z,\<2^ |̂ | <1}U
{(z^z2)eC2:3<^\<^ 1^1 <l}.
By letting e tend to 0, one sees in this simple example that the size of the
neighborhood W of K in Theorem 1 does not depend on simple data such
as the diameter of K, the distance from K to the boundary of U, etc.
2. Proof of Theorem 2.
We could simply say that the proof strictly follows [4] pages 52-
55, or [3] pages 85-91, sources which are much easier to read than [1].
However it seems better to go through the proof in some detail. In addition
to having the conclusion stated differently, our exposition presents some
minor differences from [3] or [4]: We avoid Lemma 9 in [3], page 85, and
appropriate normalizations lead to some simplification of notations.
LEMMA 2. — Let Ai,..., AN e U C C^ For a and (3 € N, there
exists a non-zero polynomial P^p in (n + 1) variables (2:1,..., 2^,w) of
degree not more than a in each variable z\,..., Zn and of degree not more
than (3 in w such that the function of n variables
F^{z) = P^(^/(^)),
ANNALES DE L'lNSTITUT FOURIER
GOOD COMPACT SETS 683
which is defined on U, satisfies
^ '^ fA^-O-a^nA)-"
for ail multi-indices I with \I\ < (TV-1/7^/?1/71) - 1.
Proof. — A simple count of dimension. There are fewer than (TV"1/71
a/?1/71)71 such multiindices, and since there are N points Aj we have fewer
than a71/? linear constraints on a vector space of polynomials of dimension
(a+W+l).
We use the notations of Theorem 2. Independently of / we now select
the points A i , . . . , AN to be used in applying Lemma 2. Let p < dist(L, bU).
Choose AI , . . . , AN G L such that
N
Lc\jB{A^p/2)^
j=i
where JE?(Aj,p/2) denotes the open ball centered at Aj and of radius p / 2 .
Normalizations. — Without loss of generality we will assume that
U C B(0,1) and that |/| < 1. (By shrinking U one can first restrict to
bounded holomorphic functions /.) A polynomial Q = Y^aIqZIwq is said to
be normalized if max \diq\ = 1. By multiplying by appropriate constant, we
can take the polynomial Pa /s in Lemma 2 to be normalized.
LEMMA 3. — With N and A i , . . . , AN as above, fix C > ̂ -{N-1^^
For each (a, (3) let Pap be a normalized polynomial as provided by
Lemma 2. There exists /3o € N such that for every a ^ 1 and every f3 ^ /3o
sup|P^/(^))|^C^1̂/^
z^L
The lemma will be used by taking an arbitrary value of C with
^-(N-1^) ̂ c <1. Notice that
s\ip\Pap(z,f{z))\ =sup|P^|.
zCL LI
Proof. — By counting the number of monomials one has the esti-
mate:
\Pa(3^,w)\ ̂  (a+ iFC^-h 1) if each \Zj\ ̂  1 and |w| ^ 1.
Applying Schwarz's Lemma to Pa/?(^ f(z}) on each ball B{Aj,p) yields
the estimate on the half balls that
sup \P^ f(z))\ ^ (a + l)^ + 1) 2-^
B(a,^/2)
TOME 50 (2000), FASCICULE 2 (special Cinquantenaire)
684 j.-p. ROSAY & E.L. STOUT
where pn is the largest integer less than TV-1/71^1/71 - 1. For a ^ 1 and
f3 ^ A), 0o large enough,
(a+l)71^^-^ ^C^17",
whence the lemma.
LEMMA 4. — There exists 7 > 0 such that for every a e N and for
every normalized polynomial Q in C" of degree at most a in each variable
and for every t C (0,1)
\{zeB^l):\Q{z)\<ta}\^———.
logf
(The above notation is for the measure of the set denned by \Q(z)\ <
r.)
The important features are: ^-r does not depend on the degree and
can be made arbitrarily small by appropriate choice of t.
This lemma is given in [3], p. 88; the proof we give, which is included
for the convenience of the reader, is a minor adaptation of the proof given
there.
Proof. — Set
F,=^eB(0,l) : |Q(^) |^^}.
By simply counting the number of monomials and using the rough estimate
that 1^1 ^ 21-7!, one gets that for z e B(0, 2)
|0(^)|^ (a +l)n2na.
If Q(z) = ̂  a j z ' 7 , let K = ( A ; i , . . . , kn) be such that |a^| = 1. We have
OK = Vn^ I Q (e^ . ^) e-^-^) ̂  . . . ̂
J \Vn ^/nj 27T 27T
so
max |<9(^)| ^ ———.
^5(0,1)"' / 1 (yn)^
Fix ^° e B(0,l) such that |Q(^)| ^ ^y. As log [Q| is subharmonic, the
mean value property on the ball B{z°, 1) gives
''̂ whyi/,,.,,,togM-
ANNALES DE L'lNSTITUT FOURIER
GOOD COMPACT SETS 685
The estimates given above then yield
TLOL 1 r
- -^ logn <; log((« + 1)-2"°) + ̂ ^ ̂  log |0|
,lo^+in-]-^»lo4
Thus
|B(0,l)|(nl^g±D+nlog2+tlogn)
I^t| ^ ———-———————_——-—-————————-
logf
As log(^ ^ 1, we can take
7=n|B(0,l)l('l+log2+ l^|ny
Proof of Theorem 2. — We use the notations of Lemmas 2, 3 and 4.
Fix e > 0 and let t > 0 be small enough that -^ ^ e. In Lemma 2,
fix C < 1. Fix f3, (3 ^ A), such that C^1771 < t. For each a ̂  1, let P^ as
in Lemma 3. Write
ft
P^=^Q,W.
j=o
At least one of the polynomials Qj must be a normalized polynomial in the
n variables z^ , . . . , Zn of degree not more than a in each variable.
Thus the set
S^ = [z e B(0,1) C C71 : max,|Q^)| ^ ̂ }
has measure at most e.
If z € B(0,1) \ S^ we set
P'2 n = ^^(Q
Qpu max,|0,(^)r
Notice that if z = (z i , . . . , Zn), then w ̂  P^(^i,..., Zn, w) is a normalized
polynomial of degree not more than /3 in the variable w.
If z e 5(0,1) CC71,
{• either z e 5a, an exceptional set of measure not more than e9 or sup^JP^I^^)'.
TOME 50 (2000), FASCICULE 2 (special Cinquantenaire)
686 J.-P. ROSAY & E.L. STOUT
^l/nRemember that —^— < 1.
The set of z which belong to all but finitely many Sa has measure
not more than e; this is the exceptional set E^ in the theorem.
If there are infinitely many a such that z $? 5o,, by passing to a
subsequence we get a sequence of polynomials Rj in n + 1 variables such
that
• sup^ |^-|—>0
J-^OO
• Gj(w) = Rj(z,w) is a normalized polynomial of degree
not more than a, in one variable, and the sequence (Gj)
converges to a normalized polynomial G.
Of course the sequence Rj (thus Gj) depends on z.
By the very definition of polynomial hull we immediately get that if
(z^w) € Li, then G(w) = 0. So there are at most (3 points in the fiber of
Li above z. (Here f3 is the k in the statement of the Theorem).
Remarks. — 1) In Theorem 2 it is not true that L\ is necessarily
finitely sheeted over C71, so that the exceptional set Ee is, in general,
nonempty. This is illustrated by the following simple example. Let L =
{(e^.Ae^) € C ^ I A I ^ 1}. The function (^1,^2) = f- is holomorphic in
a neighborhood of L. Here Li = {(e^,Ae^,A) € cC|A| ^ 1}. For every
A € C, such that |A| ^ 1, (0,0, A) G Z/i. So the fiber of Li above (0,0) is
the unit disk.
2) In an arbitrary complex manifold, good compact sets are defined
simply by replacing polynomials by global holomorphic functions. In Stein
manifolds all compact sets are good compact sets, but there are examples
of complex manifolds where even a compact set consisting of a single point
fails to be a good compact set. (See [6].) [6] contains also a related example
of indecomposable analytic functional, an analytic functional carried by a
compact set not reduced to a point, that cannot be decomposed into the
sum of analytic functionals carried by smaller compact sets.
ANNALES DE L'lNSTITUT FOURIER
GOOD COMPACT SETS 687
BIBLIOGRAPHY
[1] E. BISHOP, Holomorphic completions, analytic continuations, and the interpolation
of semi-norms, Ann. Math., 78 (1963), 468-500.
[2] J.E. BJORK, Every compact set in C71 is a good compact set, Ann. Inst. Fourier,
Grenoble, 20-1 (1970), 493-498.
[3] R.C. GUNNING, Introduction to Holomorphic Functions of Several Complex Vari-
ables, vol. I, Wadsworth and Brooks-Cole, Belmont, 1990.
[4] R.C. GUNNING and H. ROSSI, Analytic Functions of Several Complex Variables,
Prentice-Hall, Englewood Cliffs, 1965.
[5] A. MARTINEAU, Sur les fonctionelles analytiques et la transformation de Fourier-
Borel, J. Analyse Math., XI (1963), 1-164. (Also contained in the (Euvres of
Martineau).
[6] J.-P. ROSAY and E.L. STOUT, Strong boundary values, analytic functionals and
nonlinear Paley-Wiener theory, to appear.
[7] W.R. ZAME, Algebras of analytic germs, Trans. Amer. Math. Soc., 174 (1972),
275-288.
Manuscrit recu Ie 23 juin 1999.
J.-P. ROSAY,
University of Wisconsin
Department of Mathematics
Madison WI 53706 (USA).
jrosay@math.wisc.edu
E.L. STOUT,
University of Washington
Department of Mathematics
Seattle WA 98195 (USA).
stout@math.washington.edu
TOME 50 (2000), FASCICULE 2 (special Cinquantenaire)


An approximation theorem related to good compact sets in the sense of Martineau

Dal'bo, Françoise

Topologie du feuilletage fortement stable

Ghys, Étienne

Rebelo, Julio C.

ANNALES DE L’INSTITUT FOURIER
ÉTIENNE GHYS
JULIO C. REBELO
Singularités des flots holomorphes. II
Annales de l’institut Fourier, tome 50, no 3 (2000), p. 1019-1020
<http://www.numdam.org/item?id=AIF_2000__50_3_1019_0>
© Annales de l’institut Fourier, 2000, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
50, 3 (2000), 1019-1020
ERRATUM
SINGULARITÉS DES FLOTS HOLOMORPHES. II
par E. GHYS et J. C. REBELO
Article paru dans le tome 47 (1997), fascicule 4, pp. 1117-1174
Le théorème G (page 1118) contient une liste des champs de vec-
teurs homogènes dans C2 qui sont "semi-complets". Cette liste n'est pas
complète : il faut modifier légèrement le cas 4 dans l'énoncé de ce théorème
en le remplaçant par :
4. X1^ == f(x,y)(mx9/0x - nyO/Oy) où m,n sont des entiers strictement
positifs et f est un polynôme homogène. De plus f est de la forme
f(x, y) = x ' y 3 avec mi - nj = -1,0 ou 1, ou encore f{x, y) = (xy)1^ - y)
avec m = n = 1 et k = 21 + 2.
L'omission est située dans le lemme 2.14 où il s'agit de trouver les
couples d'entiers strictement positifs (m,n) et les polynômes homogènes
f(x,y) tels que, pour tout (.TI,^), la fonction zf{x^zrn,X2Z~n) de la
variable z est une combinaison linéaire à coefficients constants de 1, z et z2
(et non pas, comme il est affirmé incorrectement, un multiple constant de
1, z ou z2). Il est facile de vérifier que les seuls / qui conviennent sont ceux
signalés plus haut. Le dernier, f(x, y) = (rq/)^-1^ - y) (pour m = n = 1),
n'apparaissait pas dans la version publiée du théorème C.
Ceci n'a aucune conséquence sur le théorème A (et donc sur le
théorème B). Il s'agit en effet de s'assurer qu'un champ semi-complet X à
singularité isolée ne peut avoir un 2-jet de la forme (x - y) (xQ/9x - y 9 / 9 y ) .
L'éclatement (semi-complet) X d'un tel champ s'annulerait sur le diviseur
exceptionnel et le feuilletage associé présenterait un nœud-col dont la
1020 E. GHYS, J.-C. REBELO
séparatrice forte serait contenue dans le diviseur, en contradiction avec
le lemme 3.7.
Nous remercions A. Guillot de nous avoir signalé cette omission.
E. GHYS et J. C. REBELO,
École Normale Supérieure de Lyon
UMPA
46 allée d'Italie
69364 LYON CEDEX 07 (France).
ghys@umpa.ens-lyon.fr
ANNALES DE L'INSTITUT FOURIER


Singularités des flots holomorphes. II

Sauloy, Jacques

Systèmes aux $q$-différences singuliers réguliers : classification, matrice de connexion et monodromie

Cassaigne, Julien

Ferenczi, Sébastien

Zamboni, Luca Q.

Imbalances in Arnoux-Rauzy sequences

Böcherer, Siegfried

Schmidt, Claus-Günther

$p$-adic measures attached to Siegel modular forms

Amrane, Samir Ait

Sur le schéma de Hilbert des courbes gauches de degré $d$ et genre $g=(d-3)(d-4)/2$

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$p$ -adic measures attached to Siegel modular forms

Sur le schéma de Hilbert des courbes gauches de degré $d$ et genre $g=(d-3)(d-4)/2$

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