Laura J Blair
email: lblair@health.usf.edu

Olivier Genest
email: ogenest@imm.cnrs.fr

Mehdi Mollapour
email: mollapom@upstate.edu

The multiple facets of the Hsp90 machine

The Ninth International Conference on the Hsp90 Chaperone Machine concluded in October 2018, in Leysin, Switzerland. The program highlighted findings in various areas, including integrated insight into molecular mechanism of Hsp90, cochaperones, and clients' structure and function.

). Here, we distill and summarize their findings.

Hsp90 Dynamics

Carefully orchestrated conformational changes in Hsp90 are essential for its association with cochaperones and client proteins. David Agard (University of California, San Francisco, California, USA) provided the first atomic details for the Hsp90/Hsp70/HOP/glucocorticoid receptor (GR) client-loading complex. Unexpectedly, this heterocomplex contains two Hsp70s both in the ADP-bound state, one of which appears to be involved in the delivery of GR, while the other supports the cochaperone HOP. GR is largely intact, but like in the Hsp90:Cdc37:kinase complex [START_REF] Verba | How Hsp90 and Cdc37 Lubricate Kinase Molecular Switches[END_REF] , part of GR is unfolded, and threaded through the lumen of a partially closed Hsp90 dimer, where it interacts with a newly revealed client binding site on HOP. Together this suggests a common mechanism for Hsp90:client recognition.

Stefan Rüdiger (Utrecht University, the Netherlands) revealed that Hsp70 binds clients through highly hydrophobic regions that provide protection from misfolding. Subsequently, Hsp90 breaks this interaction and allows clients to self-fold into a native state. Cochaperones are not necessary for folding, but mainly work to slow this process. Thorsten Hugel (University of Freiburg, Germany) described cooperation between nucleotide and the two amino-terminal ATPbinding pockets in an Hsp90 dimer using multicolor, single-molecule Förster resonance energy transfer (FRET). This novel technique adds an additional dimension that allowed for the discovery that ATP and Aha1 independently, but synergistically, promote closing of the nucleotide pocket, but work antagonistically to affect subsequent reopening [START_REF] Götz | Using Three-color Single-molecule FRET to Study the Correlation of Protein Interactions[END_REF] . Katarzyna Tych from Matthias Rief's Lab (Technische Universität München (TUM), Munich, Germany) described the dynamics of Hsp90 carboxy-terminal dimerization using single molecule optical tweezers. Interestingly, Hsp90 carboxy-terminal association has three dissociation rates that are controlled by the presence of ATP, which stabilizes this interaction by eliminating the weakest interaction state. Vinay Dahiya from Johannes Buchner' group (TUM, Munich, Germany) described the chaperoning mechanism of Hsp70 and Hsp90 for "the guardian of the genome," p53. Hsp70 together with Hsp40 unfolds and inactivates p53. The Hsp70 NEF, Bag1 supports the release of p53 from Hsp70 and, in coordination with HOP, Hsp90 and ATP promotes the folding of p53 and restores its DNA binding activity. Shannon Doyle from Sue Wickner's Lab (NCI, NIH, Bethesda, Maryland, USA) showed that Hsp90 and Hsp70 directly interact in both Escherichia coli (E. coli) and yeast. The region of interaction on Hsp90 involves residues in the middledomain that also interact with several cochaperones and clients, while the interaction region on Hsp70 utilizes residues in the J-protein binding region of Hsp70 5 .

Hsp90 Phosphorylation and Regulation

Hsp90 and its cochaperones are subject to post-translational modifications (PTMs) including phosphorylation. Ioannis Gelis (University of South Florida, Tampa, Florida, USA) demonstrated that during the chaperone cycle, Hsp90 phosphorylation occurs in a cochaperone regulated manner. This is exemplified by the Hsp90/Cdc37 heterocomplex, in which the phosphorylation of Cdc37 primes Hsp90 for phosphorylation of tyrosine 197, and controls disassembly of the client recruitment complex. Thus, Hsp90 cochaperone complexes are highly dynamic and highly regulated at many steps throughout the chaperone cycle [START_REF] Bachman | Phosphorylation induced cochaperone unfolding promotes kinase recruitment and client class-specific Hsp90 phosphorylation[END_REF] .

Matthias Mayer (Universität Heidelberg, Germany) revisited previous works on PTMs of Hsp90 and how they fine tune Hsp90 function. His data based on yeast growth assays revealed that some phosphomimetic mutations of Hsp90 grow more efficiently than the wild-type Hsp90 and more potently activate steroid hormone receptors, apart from GR. This supports the idea that PTMs can tune Hsp90 specificity for chaperoning specific clients, even those that evolved from the same gene.

Structure and function of Hsp90 Orthologs

Dan Gewirth (Hauptman-Woodward Institute, Buffalo, New York, USA) presented data on the structure of the Pre-amino (Pre-N) domain of Grp94, the endoplasmic reticulum (ER) Hsp90, along with functional data showing its role in client protein maturation. The Pre-N domain regulates the ATPase activity of the chaperone and appears to functionally substitute for cochaperones of Hsp90, which are missing in the ER [START_REF] Huck | Structural and Functional Analysis of GRP94 in the Closed State Reveals an Essential Role for the Pre-N Domain and a Potential Client-Binding Site[END_REF] .

Olivier Genest (CNRS, Aix Marseille University, France) showed that Hsp90 is essential under heat stress in the aquatic bacterium Shewanella oneidensis [START_REF] Honoré | Hsp90 Is Essential under Heat Stress in the Bacterium Shewanella oneidensis[END_REF] . He found that TilS, a tRNA modifier, is a client of Hsp90 and that an interplay takes place between folding by Hsp90 and degradation by the protease HslUV to finely regulate the level of TilS.

Modulation of Hsp90 by Cochaperones

The chaperone function of Hsp90 is tightly regulated by cochaperones. Pierre Goloubinoff (University of Lausanne, Switzerland) proposed Hsp70 as the major cochaperone of Hsp90.

Deletion studies in E. coli demonstrate that, unlike the deletion of Hsp70/Hsp40, which upregulated many chaperones and proteases while suppressing metabolic and respiratory enzymes, deletion of E. coli Hsp90 did not dramatically impact protein levels. This work also revealed that Hsp90 promotes the degradation of aggregation-prone clients of Hsp70/Hsp40 through the HsIUV protease. Kaushik Bhattacharya from Didier Picard's group (University of Geneva, Geneva, Switzerland) addressed the question of the importance of the HOP protein in eukaryotes [START_REF] Bhattacharya | Luminescence resonance energy transfer between genetically encoded donor and acceptor for protein-protein interaction studies in the molecular chaperone HSP70/HSP90 complexes[END_REF] . He found that the complex Hsp70-HOP-Hsp90 physically interacted with the proteasome. Although in the absence of HOP the proteasome activity is reduced, he showed that proteostasis is maintained by a "super-chaperone" complex minimally consisting of Hsp90, Hsp70, a J-protein and a nucleotide exchange factor, resembling the chaperone complex found in prokaryotes.

Michael Reidy from Dan Masison's group (NIDDK, NIH, Bethesda, Maryland, USA) reported that Sti1 (HOP) has two distinct functions for yeast Hsp90. The first function is to connect Hsp70 to Hsp90 by binding directly to both chaperones. The second function is to facilitate transfer of clients from Hsp70 to Hsp90 by priming Hsp90 for client loading.

The R2TP complex is a cochaperone of Hsp90 that includes RuvBL1 and RuvBL2, two AAA+ proteins involved in cancer progression [START_REF] Houry | The PAQosome, an R2TP-Based Chaperone for Quaternary Structure Formation[END_REF] . Walid Houry (University of Toronto, Toronto, Canada) developed a screen to find inhibitors of the RuvBL2 protein. He successfully identified a compound that inhibits RuvBL2. Chris Prodromou (University of Sussex, Brighton, UK) described the 3D structure of the yeast and human R2TP complex that was recently solved using X-ray crystallography and single-particle EM [START_REF] Rivera-Calzada | The Structure of the R2TP Complex Defines a Platform for Recruiting Diverse Client Proteins to the HSP90 Molecular Chaperone System[END_REF] . These structures revealed the importance of the Hsp90 cochaperone complex as a hexamer-ring and that in yeast, binding of a single Tah1 coupled to Pih1 could promote Rvb1p-Rvb2p ATPase activity. Philippe Meyer (Sorbonne Université, CNRS, Paris, France) presented the crystal structure of the human RPAP3 in complex with PIH1D1, important cochaperones in the R2TP complex that regulate Hsp90 activity. This work revealed that the TPR2 domain in RPAP3 recruits Hsp90, which stimulates Hsp90 ATPase activity in synergy with Aha1. This stimulation activity is abolished if the aminoterminus of RPAP3 is deleted.

Markus Zweckstetter (DZNE, Göttingen, Germany) described the 3D structure of Hsp90 in complex with misfolded transthyretin, which was similar to the binding previously described for tau [START_REF] Oroz | Structure and pro-toxic mechanism of the human Hsp90/PPIase/Tau complex[END_REF] . The impact of cochaperones on Hsp90 structure was also described using the example of FKBP51, which binds to all three domains of Hsp90 with varying affinities and results in reduced ATPase activity by stabilizing an open conformation of the amino-terminal domain.

When bound to Hsp90, the catalytic site of FKBP51 remains accessible supporting the idea that Hsp90 likely acts as a scaffold to support the functional interaction of misfolded proteins with cochaperones. Mehdi Mollapour (SUNY Upstate Medical University, Syracuse, New York, USA) described the cooperative function of new cochaperones, Tsc1 and FNIP1/2. They work cooperatively to decelerate Hsp90 chaperone cycle and play a role in the chaperoning of both kinase and non-kinase clients [START_REF] Sager | The mTOR Independent Function of Tsc1 and FNIPs[END_REF] . He also showed how phosphorylation, SUMOylation and ubiquitination of the cochaperone Protein Phosphatase-5 (PP5) regulate its activation, substrate binding and degradation in cancer.

Moonlighting Functions of Hsp90

Several newly emerged moonlighting functions of Hsp90 at the nucleus and plasma membrane were revealed at this meeting. Brian Freeman (University of Illinois, Urbana, Illinois, USA), using an elegant single cell approach with a fluorescently marked DNA locus, demonstrated how Hsp90 is involved in chromosome motion within interphase cells. He showed that Hsp90 and p23 keep ARP-containing chromatin remodelers in a dynamic state allowing them to interact with target gene promoters, in turn directing actin polymer formation and chromosome motion.

Gergely Lukacs (McGill University, Montreal, Canada) showed that both eukaryotic (Hsc70 and Hsp90) and prokaryotic chaperone (DnaK) systems can reshape the conformational energetics of mutant CFTR channel final fold towards that of the wild-type at the single molecule level and in cells. This mechanism has implications in the regulation of metastable ABC-transporters and other membrane proteins. Dragana Vidovic, from Ineke Braakman's group (Utrecht University, The Netherlands) reported that Hsp90 is required for a crucial step in folding of CFTR. More specifically Hsp90 is involved in folding of the CFTR nucleotidebinding domain. Wild-type CFTR in the presence of compromised Hsp90 function is folded like the pathogenic mutant CFTR ∆F508. Michael Heider and Vanesa Fernandez-Saiz from Florian Bassermann's Lab (TUM, Munich, Germany) described Cereblon (CRBN), the target of immunomodulatory drugs (IMiDs), as a novel regulator of the Hsp90-Aha1 axis that mediates maturation of transmembrane proteins such as CFTR. They also showed that Hsp90-CRBN interaction was abrogated upon IMiD-treatment, thus explaining CRBN-dependent IMiD induced destabilization of its transmembrane clients.

Shiran Dror from Anat Ben-Zvi's group (Ben-Gurion University of the Negev, Beer Sheva, Israel) presented his work on the role of Hsp90 in myosin assembly in C. elegans. He showed that knockout or overproduction of Hsp90 and some of its cochaperones resulted in loss of motility phenotypes due to myosin filament disorganization. Looking at chaperone localization patterns in the sarcomere, he proposed that Hsp90 machinery is important for regulating myosin translocation across the sarcomere allowing the formation of proper filaments of myosin. Yu-Chun Wang from Patrik Verstreken's lab (VIB, Leuven, Belgium), showed a previously unexpected role of Hsp90 in membrane remodeling. In vitro and in vivo evidence indicated that Hsp90 interacts with membranes via an amphipathic helix, deforms the membranes and allows exosome release. This activity can be inhibited by Hsp90 inhibitors [START_REF] Lauwers | Hsp90 Mediates Membrane Deformation and Exosome Release[END_REF] .

Christine Queitsch (University of Washington, Seattle, Washington, USA) described that the effects of Hsp90 on the evolutionary rate of protein kinases is comparable to effects of gene expression and protein interactions. She showed that a single mutation can render a non-client transcription factor Hsp90-dependent and that Hsp90-dependent de novo mutations do not account for most Hsp90-dependent phenotypes.

Hsp90 in Maladies

Hsp90 is involved in various maladies including cancer, neurodegenerative diseases, and pathogenic infections. Laura Blair (University of South Florida, Tampa, Florida, USA) demonstrated that Aha1 increases tau aggregation. In contrast, cyclophilin 40 (CyP40) disrupts tau fibrils reducing neurotoxicity in vivo [START_REF] Baker | Human cyclophilin 40 unravels neurotoxic amyloids[END_REF] . These findings provide strong rationale to develop therapeutics for tauopathies by targeting molecular chaperones. Aaron Voigt (RTWH Aachen University, Germany) demonstrated that mitochondrial Hsp90 TRAP1 is an important player in Parkinson's disease (PD) and his data suggest that enhancing TRAP1 abundance or activity in neurons might be an avenue for future PD therapies.

Oliver Krämer (Johannes Gutenberg University Mainz, Germany) presented his work on the involvement of Hsp90 and the histone deacetylase 6 (HDAC6) in acute myeloid leukemia (AML) cells. He showed that a combination of inhibitors directed towards HDAC6 and Hsp90 triggered apoptosis of AML cells, providing new promising ways to treat AML. Ramona Schulz-Heddergott (University Medical Center of Göttingen, Germany) described that stabilization of a cancer-relevant Hsp90 client p53-R248Q contributes to STAT3 hyper-activity. Vice versa, Hsp90 inhibition by 17AAG leads to degradation of the p53 mutant and thus, down-regulation of p-STAT3. This data suggests that the cancer-relevant Hsp90 client p53 mutants represent an actionable drug target for treatment with Hsp90 inhibitors.

Stephanie Diezmann (University of Bristol, Bristol, UK) described how Hsp90 and lack of the cochaperone Sti1 (HOP) enables phenotypic variation via loss-of-heterozygosity and aneuploidy in the ameiotic human fungal pathogen Candida albicans. She proposed the environmentally responsive chaperone Hsp90 as a novel mechanism for the creation of genetic diversity in C. albicans. Harriet Mok from Jason Mercer's group (University College London, UK) highlighted the importance of Hsp90 in the life cycle of a poxvirus family member, vaccinia virus (VACV). Using microscopy and virus-specific assays, she showed that multiple isoforms of Hsp90 are required during late stages of vaccinia infection including successful genome release into the host cytoplasm and replication.

Joachim Clos (Bernhard Nocht Institute for Tropical Medicine, Hamburg, Germany) discussed the translatome changes in Leishmania donovani following Hsp90 inhibition. This protozoan parasite showed increased synthesis of proteins that are involved in oxidative stress protection, proteolysis, chromatin assembly, and chaperoning following Hsp90 inhibition. These results underscore the importance of Hsp90 in the control of the parasite's life cycle.

Hsp90 in the Extracellular Environment

The extracellular molecular chaperone Hsp90 (eHsp90, released or surface-bound) is responsible for chaperoning clients outside the cell. Wei Li (University of Southern California, Los Angeles, California, USA) provided a remarkable overview of the roles of extracellular Hsp90α in repairing injured tissues and in supporting tumorigenesis. During wound healing, secreted Hsp90α interacts with the extracellular domain of a receptor (LRP-1) that is stabilized by Hsp90β inside the cell. Hsp90α protects cells from hypoxia-triggered apoptosis and promotes cell motility to close the wound. Surprisingly, the activity of Hsp90α is independent of its ATPase activity and can be fulfilled by a short region of Hsp90α of only about 115 amino acids, called fragment-5 [START_REF] Cheng | A fragment of secreted Hsp90alpha carries properties that enable it to accelerate effectively both acute and diabetic wound healing in mice[END_REF] . These properties of secreted Hsp90α have been taken advantage of by tumor cells to gain invasion and metastasis, representing an alternative target for anti-tumor therapeutics. Natasha Boel from Adrienne Edkins' group (Rhodes University, Grahamstown, South Africa) investigated the role of Hsp90 in the dynamics of the fibronectin (FN) matrix. She showed that Hsp90 interacts with FN, and that inhibition of Hsp90 by novobiocin resulted in FN turnover via a LRP-1 receptor mediated response [START_REF] Boel | LRP1 is required for novobiocin-mediated fibronectin turnover[END_REF] .

Patricija van Oosten-Hawle (University of Leeds, Leeds, UK) explained how a proteotoxic stress perceived in one tissue induces a response in other tissues leading to molecular chaperone activation via a pathway termed transcellular chaperone signaling (TCS). Using C. elegans, she showed that the zinc finger transcription factor PQM-1 is involved in mediating TCS [START_REF] O'brien | A PQM-1-Mediated Response Triggers Transcellular Chaperone Signaling and Regulates Organismal Proteostasis[END_REF] . Importantly, she found that activating Hsp90 by TCS reduced the formation of toxic amyloid beta aggregates.

Small molecule inhibitors of Hsp90

Hsp90 inhibitors that target the ATP binding pocket in the amino-terminal domain are currently evaluated in cancer patients. Brian Blagg (University of Notre Dame, Notre Dame, Indiana, USA) presented the design and development of two new scaffolds that selectively inhibit the Hsp90ß isoform [START_REF] Khandelwal | Structure-guided design of an Hsp90beta N-terminal isoformselective inhibitor[END_REF] . As a result of these studies, new isoform-dependent substrates were identified that can be selectively modulated via inhibition of Hsp90ß. In addition, it was observed that selective inhibition of Hsp90ß induces the degradation of HSF1, and thus, there is no induction of the heat shock response. Initial studies suggest that Hsp90ß-selective inhibition may overcome some of the obstacles encountered with the pan-Hsp90 inhibitors that have struggled in clinical trials. Timothy Haystead (Duke University, Durham, North Carolina, USA) presented data on tethered inhibitors of Hsp90 amino-terminal domain causing aggregation and reinternalization of the Hsp90 protein such that it accumulates intracellularly at high concentration. This observation led to development of a series of imaging agents for the detection of early metastatic disease as well as imaging of inflammatory responses to immune challenge.

Understanding Hsp90 Function Through Artificial Intelligence

William E. Balch (The Scripps Research Institute, La Jolla, California, USA) used a new Gaussian Process-based machine learning approach referred to as Variation Spatial Profiling (VSP) to show that diversity at the population level is sensitive to the buffering capacity of the proteostasis program. Based on Spatial CoVariance (SCV), genetic diversity contributing to misfolding disease found in the population can be displayed as high-dimensional phenotype landscapes. Application of these technologies reveal a map of the specific sequence-to-function-to-structure features of the client protein fold that are responsive to Hsp70-Hsp90 chaperone/cochaperone management at molecular resolution [START_REF] Wang | Bridging Genomics to Phenomics at Atomic Resolution through Variation Spatial Profiling[END_REF] . Gennady Verkhivker (Chapman University School of Pharmacy, Irvine, California, USA) described the step-wise fashion by which the Hsp90-Cdc37 heterocomplex recognizes client kinases through differential stabilization of kinase lobes. This work revealed that Hsp90 binds after Cdc37 and helps shed light on unique dynamics signatures of protein kinase clients/nonclients that dictate chaperone addiction and explain divergences in the regulatory mechanisms among structurally similar kinases.

Recognition of Colleagues in the Hsp90 Field

For the past sixteen years two leaders in the field, Didier Picard (University of Geneva, Switzerland) and Johannes Buchner (TUM, Munich, Germany), have organized the Hsp90 conference biannually. At this 9 th International Conference, they were recognized for their valuable contribution and outstanding service to the Hsp90 field and community. They were both acknowledged and thanked for their continuous support of students, post-doctoral fellows and new group leaders (Figure 1).

Figure legend Figure 1 .

 1 Figure legend Figure 1. Above-Participants of the Ninth International Conference on the Hsp90 Chaperone Machine. Below-Brian Freeman (far left) and Mehdi Mollapour (far right) presented recognition trophies to the meeting organizers, Johannes Buchner, Didier Picard (middle). Photos by Abhinav Joshi.


Daniel Barlet

On symmetric partial differential operators

Let s 1 , . . . , s k be the elementary symmetric functions of the complex variables x 1 , . . . , x k . We say that

We give an explicit finite family of second order differential operators in the Weyl algebra

which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (-1) h .s h .z k-h = 0.

Introduction

Let x 1 , . . . , x k be the roots of the monic polynomial P s (z) := k h=0 (-1) h .σ h .z k-h . For any entire function f : C → C define the trace T (f ) of f as the entire function on C k define by T (f )(σ) := k j=1 f (x j ). The aim of this paper is to construct an explicit finite set of second order differential operators in the Weyl algebra

W 2 := C[σ 1 , . . . , σ k ] ∂ σ 1 , . . . , ∂ σ k
which annihilate any trace functions and such that they give a characterization of entire functions of this type. In fact we shall prove more: our explicit finite set of second order differential operators will generate the left ideal in W 2 of all differential operators killing all trace functions. As it is clear that a trace function view as a function on x 1 , . . . , x k is killed by any elementary symmetric functions of order at least two of ∂ x 1 , . . . , ∂ x k our first approach is to write these symmetric differential operators as elements in W 2 . But one can see directly from the cases k = 2, 3 that this is not obvious and even the question of the existence of such operators is not clear. So we are lead to prove an analogous result to the usual symmetric functions theorem for symmetric differential operators (see the theorem 3.2.1). After this result which gives the existence, it appears that explicit computation of the elements Σ h ∈ W 2 corresponding to the elementary symmetric functions of order at least two of ∂ x 1 , . . . , ∂ x k is out of the range of what can be computed explicitly (see the formula of Σ 3 for k = 3). Then, using an integral formula to compute trace functions in the spirit of Lisbon integrals, see , I find an explicit finite family of second order differential operators in W 2 which annihilates any trace function. Then the aim of the second part of the paper is devoted to the proof that any entire function of σ 1 , . . . , σ k which is killed by this family is a trace function.

To prove this result we have to understand how the symbols behave via the theorem of symmetric differential operators which is proved in section 3, in order to describe the characteristic variety of the system given by our explicit family of second order differential operators and to compare with the characteristic variety of the left ideal in W 2 of all differential operators which are killing trace functions. This allows us to show that our second order differential operators generate the left ideal in W 2 of all differential operators which kill all the trace functions. An easy corollary of this result is to obtain explicit generators of a left ideal in W 2 of the differential operators which annihilate the branches of the multivalued root of the universal equation of degree k : z k + k h=1 (-1) h .σ h .z k-h = 0. This system is holonomic and its 0-th sheaf of solutions is the C -constructible sheaf generates by local branches of the multivalued root of the universal equation of degree k. An other interesting consequence of our result is that we can also find an explicit finite family of second order differential operators in W 2 which generates the ideal of differential operators in W 2 which kills all the "trace forms" defined by

T f (z).dz ∧ dσ 1 ∧ • • • ∧ dσ k-1 (σ) := k j=1 f (x j ) P σ (x j ) .dσ 1 ∧ • • • ∧ dσ k
where f ∈ O(C) and where σ 1 , . . . , σ k are the symetric functions of x 1 , . . . , x k . We conclude this article by a description of symmetric derivations.

A simple problem 2.1

To each entire function f : C → C we associate the entire function T (f ) on C k defined by T (f )(x 1 , . . . , x k ) := k j=1 f (x j ). We shall call it the trace of f . Of course, this is a symmetric function in x 1 , . . . , x k and so the trace of f may be seen as an entire function in the variables σ 1 , . . . , σ k defined by the elementary symmetric functions of x 1 , . . . , x k .

Example. Any Newton symmetric function N m (x 1 , . . . , x k ), m ∈ N, considered as a polynomial in σ 1 , . . . , σ k is a trace function.

If the Taylor expansion of f at z = 0 is given by f (z) = ∞ m=0 a m .z m then we have T (f )(σ) = ∞ m=0 a m .N m (σ) where N m (σ) is the m-th Newton symmetric polynomial corresponding to the roots of the monic polynomial

P σ (z) := k j=1 (z -x j ) = k h=0 (-1) h .σ h .z k-h
with the convention σ 0 ≡ 1.

Remark. The uniform convergence on any compact set of C k for the expansion of T (f )(σ) in quasi-homogeneous polynomials given above, allows to apply any partial derivative term-wise to such an expansion. So a trace function is an uniform limit on compact sets of linear combinations (with constant coefficients) of the Newton polynomials.

The following lemma gives a characterization for an entire symmetric function on C k to be a trace function as solutions of a very simple order two differential system.

Lemma 2.1.1 For an entire function F (x 1 , . . . , x k ) which is symmetric in x 1 , . . . , x k , a necessary and sufficient condition for the existence of an entire function f on C such that F = T (f ) is given by the following differential system

∂ 2 F ∂x i ∂x j = 0 ∀(i, j) ∈ [1, k] 2 with i = j.
(1)

Proof. The necessary assumption is obvious. So consider a symmetric entire function solution of the system (1). The partial derivative ∂F ∂x 1 does not depends on x 2 , . . . , x k , so we have ∂F ∂x 1 (x) = g(x 1 ) for some entire function g on C. By symmetry, we conclude that

∂F ∂x j (x) = g(x j ) for each j ∈ [1, k]. If f is a primitive of g we conclude that F -T (f ) is a constant function on C k ,
so adding a suitable constant to our initial choice for f allows to conclude.

Remark.

It is an easy exercice to prove that the ideal generated by the symbols of the oprerators ∂ 2 ∂x i ∂x j for i = j is reduced. This implies that the left ideal of C[x 1 , . . . , x k ] ∂ ∂x 1 , . . . , ∂ ∂x k which is the annihilator of any trace function is generated by the operators ∂ 2 ∂x i ∂x j for i = j.

2.2

Now we want to obtain an analogous characterization of trace functions but with a system given by symmetric differential operators.

So introduce in the Weyl algebra

W 1 := C[x 1 , . . . , x k ] ∂ ∂x 1 , . . . , ∂ ∂x k the elementary symmetric "functions" S 1 , . . . , S k of the derivations ∂ ∂x j , j ∈ [1, k].
They are in the sub-algebra W S k 1 of symmetric differential operators. Then we have the following result.

Proposition 2.2.1 Let F be an entire function on C k which is symmetric in x 1 , . . . , x k . Then there exists an entire function f on C such that F = T (f ) if and only if F satisfies the system of partial differential equations

S h (F ) = 0 ∀h ∈ [2, k] (2)
The proof will use several lemmas.

Definition 2.2.2 Let F h be the sub-vector space of T k := C[x 1 , . . . , x k ] generated by the monomials of the type

x α 1 i 1 ...x α h i h where 1 ≤ i 1 < • • • < i h ≤ k and α ∈ N h . Note that F h is stable by the action of S k on T k . Lemma 2.2.3 The kernel of the differential operator S k acting on T k is F k-1 .
Proof. We shall make an induction on k ≥ 1. For k = 1 we have F k-1 = F 0 = C (by convention x α ≡ 1 for α = 0) and the assertion is clear. Let asume that k ≥ 2 and the lemma proved for k -1. As it is clear that F k-1 is contained in Ker(S k ), consider P ∈ Ker(S k ) and write

P = d p=0 a p .x p k where a p is in T k-1 for each p. Now ∂ k-1 P ∂x 1 ...∂x k-1 is independant of x k so belongs to T k-1 .
Then we obtain that S k-1 (a p ) = 0 for each p ≥ 1 and the induction hypothesis implies that a p is in

F k-2 ⊂ T k-1 for each p ≥ 1. So this implies that a p .x p k belongs to F k-1 ⊂ T k for each p ≥ 1. As, by definition, a 0 is in T k-1 ⊂ F k-1 we conclude that P is in F k-1 . Fix two integers 1 ≤ h ≤ k and consider the symmetrization map σ h,k : C[x 1 , . . . , x h ] → C[x 1 , . . . , x k ] S k given by σ h,k (P )(x 1 , . . . , x k ) := (k -h)! k! . i∈I(h,k) P (x i 1 , . . . , x i h )
where I(h, k) is the set of injections of {1, 2, . . . , h} into {1, 2, . . . , k}.

Lemma 2.2.4 For any integer h ≤ k we have

F h ∩ C[x 1 , . . . , x k ] S k = σ h,k C[x 1 , . . . , x h ] S h . Proof. This is obvious because σ h,k (P ) = σ h,k (σ h,h (P )). Lemma 2.2.5 The map σ h,k is injective on C[x 1 , . . . , x h ] S h .
Proof. We shall make an induction on h ∈ [0, k]. Assume that σ h-1,k is injective (this is obvious for h = 1) and consider P ∈ C[x 1 , . . . , x h ] S h such that σ h,k (P ) = 0. Then σ h,k (P )(x 1 , . . . , x h , x h , . . . , x h ) = and this implies that

(k -h + 1).P (x 1 , . . . , x h ) + Q(x 1 , . . . , x h ) = 0 where Q is in F h-1 ⊂ T h . Apply σ h,k to this equality gives that P belongs to F h-1 ∩ C[x 1 , . . . , x h ] S h
, and, thanks to the lemma 2.2.4 we obtain that

P = σ h-1,h ( Q) where Q is in C[x 1 , . . . , x h-1 ] S h-1 . So we have 0 = σ h,k (P ) = σ h,k (σ h-1,h ( Q) = σ h-1,k ( Q).
The induction hypothesis gives Q = 0 and so P = 0.

Proof of the proposition 2.2.1. Consider first a Q ∈ C[x 1 , . . . , x h ] S h such that S h (σ h,k (Q)) = 0. As we have h!.S h = i∈I(h,k) ∂ h ∂x i(1) . . . ∂x i(h) and ∂ h ∂x i(1) . . . ∂x i(h) Q(x j(1) . . . x j(h) ) = 0 f or i = j in I(h, k) S h
and equal to

∂ h ∂x 1 ,...,∂x h (Q) (x 1 , . . . , x h ) for i = j in I(h, k) S h
, where we consider the natural action of S h on I(h, k) by composition, we obtain

h!.S h (σ h,k (Q))(x 1 . . . x k ) = σ h,k ∂ h ∂x 1 . . . ∂x h (Q) (x 1 . . . x k ) = 0. Now the lemma 2.2.5 gives ∂ h ∂x 1 ...∂x h (Q) = 0. But the kernel of ∂ h ∂x 1 ...∂x h restricted to C[x 1 , . . . , x h ] S h is F h-1 ∩ C[x 1 , . . . , x h ] S h thanks to lemma 2.2.3, so Q is in F h-1 ∩ C[x 1 , . . . , x h ] S h . Then there exists, thanks to the lemma 2.2.4, R ∈ C[x 1 , . . . , x h-1 ] S h-1 such that Q(x 1 , . . . , x h ) = σ h-1,h (R)(x 1 , . . . , x h ) = 1 h . h j=1 R(x 1 , . . . , xj , . . . , x h ).
Then we have

σ h,k (Q) = σ h-1,k (R) in C[x 1 , . . . , x k ] S k . So we have proved that if Q ∈ C[x 1 , . . . , x h ] S h satisfies S h (σ h,k (Q)) = 0 there exists R ∈ C[x 1 , . . . , x h-1 ] S h-1 such that σ h-1,k (R) = σ h,k (Q). Note that this implies that so that S h-1 (σ h,k (Q)) = 0 implies ∂ h-1 R ∂x 1 ...∂x h-1 = 0 as σ h-1,k is injective. Then if P is solution of S h (P ) = 0 for each h ∈ [2, k] we construct by a descending induction a sequence Q h ∈ C[x 1 , . . . , x h ] S h such that P = σ h,k [Q h ]. So we find finally, thanks to the first part of this proof, a Q 1 ∈ C[x 1 ] with P (x 1 , . . . , x k ) = 1 k . k j=1 Q 1 (x j ) = T ( 1 k .Q 1 )(x 1 , . . . ,
P = Q(S 1 ) modulo (S 2 , . . . , S k ) where Q is in C[z]. Fixing ξ 2 = • • • = ξ k = 0 we obtain, as P is in I, Q(ξ 1 ) = 0
for any value of ξ 1 . So Q = 0 and P belongs to the ideal generated by S 2 , . . . , S k . Moreover this ideal is obviously prime in

C[S 1 , . . . , S k ] = C[ξ 1 , . . . , ξ k ] S k .
Remark. Note that the ideal I is reduced in C[ξ 1 , . . . , ξ k ] (but not prime).

Proof of the proposition 2.2.6. Let P be in W S k 1 and assume that P kills any Newton polynomial, is not in the left ideal generated by S 2 , . . . , S k in W S k 1 and has minimal order for these properties. Then the remark following the lemma 2.1.1 implies that P is in the left ideal of W 1 generated by the ∂ 2 ∂x i ∂x j , i = j. So the symbol of P is in I ∩ C[ξ 1 , . . . , ξ k ] S k . Using the lemma above, there exists Q in the left ideal generated by S 2 , . . . , S k in W S k 1 such that Q has the same symbol than P . So the order of P -Q is strictly less than the order of P and its kills any Newton polynomial. Then P -Q is in left ideal generated by S 2 , . . . , S k in W S k 1 . Contradiction ! So the lemma is proved.

Nevertheless we are not happy with this result because, using the symmetric function theorem, we are in fact working with an entire function F on C k with coordinates σ 1 , . . . , σ k corresponding to the elementary symmetric functions of x 1 , . . . , x k and we would like to have a system of partial differential operators in the Weyl algebra

W 2 := C[σ 1 , . . . , σ k ] ∂ ∂σ 1 , .
. . , ∂ ∂σ k characterizing the entire functions F which are trace functions.

So the problem is know to rewrite the system (2) of the proposition 2.2.1 in term of elements in W 2 . This will be the aim of the next section.

3 The symmetric function theorem for linear differential operators

It will be convenient to look at the local version in the study of trace functions. This is the aim of our first paragraph.

Notations.

• We shall denote by x := (x 1 , . . . , x k ) a point in M := C k , by σ := (σ 1 , . . . , σ k ) a point in N := Sym k (C) C k and by s : M → N the quotient map given by the elementary symmetric functions x → (σ 1 = s 1 (x), . . . , σ k = s k (x)).

• We shall denote by ∆ :

N → C the discriminant polynomial. So ∆(s(x)) = 1≤i<j≤k (x i -x j ) 2 . (∆)

Local trace functions

In this section we shall consider the hypersurface

H := {(σ, z) ∈ N × C / P σ [z] = 0}. (H) where P σ [z] := z k + k h=1 (-1) h .σ h .z k-h . It is smooth an isomorphic with C k via the map (σ, z) → (σ 1 , . . . , σ k-1 , z).
We shall denote by π : H → N and p : H → C, the natural projections. Definition 3.1.1 Let V be an open set in N and let F be a holomorphic function on V . We shall say that F is a (global) trace function on V if there exists a holomorphic function f on p(π -1 (V )) such that

F (σ) = T race(f )(σ) := Pσ[x j ]=0 f (x j ), ∀σ ∈ V (T) counting multiplicities. If the holomorphic function F on V is a trace function in a neighborhood of any point in V we shall say that F is a local trace function on V Examples. 1. For each m ∈ N the trace function of the polynomial function z → z m is the m-th Newton polynomial N m in C[σ 1 , . . . , σ k ]. 2. Let Q ∈ C[z]
and fix a point σ 0 ∈ N such that P σ 0 admits a simple root x 0 . Let V be an open neighborhood of σ 0 on which their exists a holomorphic function ϕ :

V → C such that ϕ(σ 0 ) = x 0 and P σ [ϕ(σ)] = 0 for each σ ∈ V . Then ϕ is a trace function on V . But if g is any holomorphic function on the open set ϕ(V ) ⊂ C, σ → g(ϕ(σ)) is also a trace function on V . Lemma 3.1.2 If F is a (global) trace function on the open set V ; then the holo- morphic function f on p(π -1 (V )) such that F = T race(f ) on V is unique up to a locally constant function. So when p(π -1 (V )) is connected, f is unique.
Proof. It is enough to consider a holomorphic function g on p(π -1 (V )) such that T race(g) = 0 on V and to prove that g is locally constant on

p(π -1 (V )). Fix a point σ 0 ∈ V such that ∆(σ 0 ) = 0. Then choose open discs D 1 , . . . , D k in C such that 1. D 1 , . . . , D k are disjoint.
2. Any D j contains exactly one root of P σ 0 which is its center.

Let V := q(D 1 × • • • × D k ). Then their exist holomorphic maps ϕ j : V → D j for j ∈ [1, k] such that we have for any σ ∈ V P σ [z] = k j=1 (z -ϕ j (σ)), ∀z ∈ C . Now fix x 0 2 , . . . x 0 k respectively in D 2 , . . . , D k . Then T race(g) at the point s(x 1 , x 0 2 , . . . x 0 k ) vanishes for any x 1 ∈ D 1 . This implies that g is constant on D 1 . So g is constant on each D j , j ∈ [1, k]. Then g is locally constant on p(π -1 (V )) as the complement of a hypersurface in a connected open set is connected. Corollary 3.1.3 Let V be a connected open set in N such that p(π -1 (V )
) is (connected and) simply connected. Then any local trace function F on V is a global trace function on V .

Proof. Let (V a ) a inA be an open covering of V such that for each a ∈ A their exists a holomorphic function f a on p(π -1 (V a )) such that F is the trace of f a on V a . Then by the previous lemma we know that f a -f b is a locally constant function on

p(π -1 (V a )) ∩ p(π -1 (V b )). This define a 1-cocyle on the open covering of p(π -1 (V ))
with value in the constant sheaf. As we assume that p(π -1 (V )) is connected and simply connected, up to pass to a finer covering, we may assume that their exist locally constant functions g a on p(π

-1 (V a )) with f a -f b = g a -g b on the intersection p(π -1 (V a )) ∩ p(π -1 (V b )). Let f be the global holomorphic function on p(π -1 (V )) given by f a -g a on the p(π -1 (V a )) Then the trace of f is a holomorphic function F on V and F -F is locally constant on V . So it is constant equal to some γ ∈ C on V . Then the trace of function f -γ/k is equal to F . Example. When V is the image by the quotient map s : M → N of D k where D is an open disc in C, V is connected and p(π -1 (V )) = D is simply connected. Lemma 3.1.4 Let V be a connected open set in N such that p(π -1 (V )
) is connected and simply connected. Then F ∈ O(V ) is a trace function if and only if it is a uniform limit on compact sets in V of linear combinations of Newton polynomials.

Proof. As p(π -1 (V )) is a Runge domain, any holomorphic function on it is a uniform limit on compact sets of polynomials. But the trace of a polynomial is a linear combination (with constant coefficients) of the Newton polynomials, and the result follows.

The theorem

Notations.

• Let e 1 , . . . , e k be then the standard basis of

T M M × C k and ε 1 , . . . , ε k the standard basis of T N N × C k . • Let T s : T M → s -1 (T N ) the tangent map to s. We shall identify a vector in T M,x
with a linear form of T * M,x In the same manner we shall identify a vector in T N,σ (or in s -1 (T N ) x ) with a linear form on

T * N,σ (resp. in s -1 (T * N ) x s -1 (T N ) * x )
. With this convention, the symbol of ∂ ∂x i is the vector field e i identified with the function e i on T * M (linear on the fibers), and the symbol of ∂ ∂σ h is the vector field ε h identified with the function ε h on T * N (linear on the fibers).

• The tangent map T s sends the vector (x, e i ) to the vector T s (x, e i ) ∈ T N,s(x)

T s (x, e i ) = (s(x), k h=1 ∂s h ∂x i (x).ε h ).
As we have (see the lemma 3.2.4 below)

∂s h ∂x i (x) = h-1 p=0 (-x i ) p .s h-p-1 (x) this gives, for (x, e i ) ∈ T M,x , i ∈ [1, k] T s (x, e i ) = (s(x), k h=1 h-1 p=0 (-x i ) p .s h-p-1 (x) .ε h ).
(3)

• We shall write (x, ξ), ξ := k i=1 ξ i .e i , a point in T M,x , (σ, η), η := k h=1 η h .ε * h , a point 1 in T * N,σ and also (x, s(x), η) a point in s -1 (T * N ) x . • Let now introduce for h ∈ [1, k] and σ ∈ N the polynomial Θ h (z, σ) := h-1 p=0 (-z) h-p-1 .σ p .
Then the formula (3) may be written in the following way

T s (x, ξ) = k h=1 k i=1 Θ h (x i , s(x)).ξ i .ε h
and then the cotangent map T * s : s -1 (T N ) * → T * M which is defined by the equality T * s (x, s(x), η), ξ = T s (x, ξ), η may be written

T * s (x, s(x), η) = k i=1 k h=1 Θ h (x i , s(x)).η h .e * i ( 4
)
so that the components in the basis e * 1 , . . . , e * k are the numbers k h=1 Θ h (x i , s(x)).η h .

• Let D M (resp. D N ) be the sheaf of holomorphic differential operators2 on M (resp. on N ).

• Let G ⊂ M × N be the graph of the quotient map s, and let p 1 : G → M and p 2 : G → N the natural projections. Note that p 1 is an isomorphism and that p 2 is a proper finite surjective map.

• The proof of this theorem will use the following simple lemma.

Lemma 3.2.2 We have natural isomorphisms of sheaves of O N -algebras

(p 2 ) * (p * 1 (D M )) → s * (D M ) (p 2 ) * (p * 1 (D M )) S k → s * (D M ) S k (p 2 ) * (p * 2 (D N )) S k → D N Proof.
The first isomorphism is clear as p 1 is an isomorphism and as we have the equality p -1 1 • p 2 = s. This implies the second isomorphism by S k -equivariance. To prove the last isomorphism, remark that we have

(p 2 ) * (p * 2 (D N )) (p 2 ) * (O G ) ⊗ O N D N and so : (p 2 ) * (p * 2 (D N )) S k (p 2 ) * (O G ) S k ⊗ O N D N ; as (p 2 ) * (O G ) S k O N using θ and the S k -equivariant isomorphism p 1 which gives s * (O M ) (p 2 ) * (O G ).
Proof of the theorem. The tangent map to the map p 2 may be view as a map of vector bundles T :

(p 1 ) * (T M ) → p * 2 (T N ) on G. It induces a morphism of sheaves of O G -algebras T : p * 1 (D M ) → p * 2 (D N
) as vector fields generate the algebra of differential operators. This morphism is equivariant for the action of S k on G and, thanks to the equivariance of p 2 we obtain a morphism of O N -algebras

Ξ 0 : (p 2 ) * (p * 1 (D M )) S k → (p 2 ) * (p * 2 (D N )) S k .
This gives the definition of the morphism Ξ using the last two isomorphisms of the previous lemma.

SΞ : s * (Gr • D M ) S k → Gr • D N
which sends the symbol of a non zero symmetric differential operator P ∈ s * (D M ) S k to the symbol of Ξ(P ) in GR • D N .

Proof. Remark that the morphism Ξ preserves the order of a non zero differential operator. So the morphism Ξ respects the filtrations of the sheaves s * (D M ) S k and D N by the order of the differential operators. Then the morphism SΞ is obtained from the morphism Ξ by passing to the graded algebras associated to these filtrations.

Let us finish this paragraph by the simple lemma used above. Lemma 3.2.4 Let x 1 , . . . , x k be complex numbers and let s 1 , . . . , s k their elementary symmetric functions. For j ∈ [1, k] let s h (j) the h-th symmetric function of x 1 , . . . , xj , . . . , x k with the convention s 0 = s 0 (j) = 1. Then we have

∂s h ∂x j = s h-1 (j) = h-1 q=0 s h-q-1 .(-x j ) q ∀ h ∈ [1, k]. (6)
Let S k (j) be the stabilizer of j in S k , the permutation group of {1, 2, . . . , k}. The subalgebra of S k (j)-invariant elements in C[x 1 , . . . , x k ] is equal to the sub-algebra generated by C[s 1 , . . . , s k ] and x j . Moreover this sub-algebra is a free C[s 1 , . . . , s k ]-module with basis 1, x j , . . . , x k-1 j .

Proof. The first equality in ( 6) is clear. The second equality in ( 6) is obvious for h = 1. Then assume that it is proved for h -1 ≥ 1. Using the easy equality s h (j) = s h -x j .s h-1 (j) we obtain

s h (j) = s h -x j . h-1 q=0 s h-q-1 .(-x j ) q = h q=0 s h-q .(-x j ) q .
concluding the induction. The last assertions follows immediately.

Remark. The morphism SΞ of O N -algebras is completely determined by the formula (3).

Examples

The case k = 2. For k = 2 we obtain that to ∂ 2 ∂x 1 ∂x 2 corresponds via Ξ the differential operator

Σ 2 = ∂ 2 ∂σ 2 1 + σ 1 . ∂ 2 ∂σ 1 ∂σ 2 + σ 2 . ∂ 2 ∂σ 2 2 + ∂ ∂σ 2 in W 2 .
Exercice. Check that Σ 2 kills the Newton polynomials N m (σ), ∀m ∈ N hint : use J. Varouchas formula's :

N m (x 1 , . . . , x k ) = α∈N k ,||α||=m (-1) m+|α| .m. (|α| -1)! α! .σ α
where |α| := k j=1 α j and ||α|| := k j=1 j.α j .

The case k = 3. Let Σ h = Ξ(S h ), where S h is the h-th elementary symmetric function of ∂ ∂x 1 , . . . , ∂ ∂x k . Some tiring computations give

Σ 2 = 3∂ 2 1 + (σ 2 1 + σ 2 ).∂ 2 2 + σ 1 .σ 3 ∂ 2 3 + 4σ 1 .∂ 1 ∂ 2 + 2σ 2 .∂ 1 ∂ 3 + (σ 1 .σ 2 + 3σ 3 ).∂ 2 ∂ 3 + 3∂ 2 + σ 1 .∂ 3 Σ 3 = ∂ 3 1 + (σ 1 .σ 2 -σ 3 ).∂ 3 2 + σ 2 3 .∂ 3 3 + 2σ 1 .∂ 2 1 ∂ 2 + σ 2 .∂ 2 1 ∂ 3 + + (σ 2 + σ 2 1 ).∂ 1 ∂ 2 2 + (σ 1 .σ 2 + 3σ 3 ).∂ 1 ∂ 2 ∂ 3 + σ 1 .σ 3 .∂ 1 ∂ 2 3 + + (σ 1 .σ 3 + σ 2 2 ).∂ 2 2 ∂ 3 + 2σ 2 .σ 3 .∂ 2 ∂ 2 3 + 3∂ 1 ∂ 2 + σ 1 .∂ 1 .∂ 3 + + 3σ 2 .∂ 2 ∂ 3 + 2σ 1 .∂ 2 2 + 3σ 3 .∂ 2 3 + ∂ 3
Let us check Σ 3 N 6 . As we have

N 6 = σ 6 1 -6σ 4 1 .σ 2 + 6σ 3 1 .σ 3 + 9σ 2 1 .σ 2 2 -12σ 1 .σ 2 .σ 3 -2σ 3 2 + 3σ 2 3 Σ 3 N 6 = (120σ 3 1 + (σ 1 .σ 2 -σ 3 )(-12) + 2σ 1 .(-72σ 2 1 + 36σ 2 ) + σ 2 .(36σ 1 ) + (σ 2 + σ 2 1 ).(36σ 1 )+ + (σ 1 .σ 2 + 3σ 3 ).(-12) + 3(-24σ 3 1 + 36σ 1 .σ 2 -12σ 3 ) + σ 1 .(18σ 2 1 -12σ 2 ) + 3σ 2 .(-12σ 1 )+ + 2σ 1 .(18σ 2 1 -12σ 2 ) + 3σ 3 .(6) + 6σ 3 1 -12σ 1 .σ 2 + 6σ 3
and then Σ 3 N 6 = σ 3 1 .(120 -144 + 36 -72 + 18 + 36 + 6)+ + σ 1 .σ 2 .(-144 -12 + 72 + 36 + 36 -12 + 108 -12 -36 -24 -12)+ + σ 3 .(36 + 12 -36 -36 + 18 + 6) = 0 Conclusion. The system given by Σ h (F ) = 0 for each h ∈ [2, k] which gives a characterization of the trace functions thanks to the previous results, seems out of the range of an explicit computation for k big and we shall now try to find a computable family of elements in W 2 which gives a characterization of the trace functions.

The aim of the next section will be to construct such a nice explicit family which will be our candidate for such a characterization.

4 Some second order PDE killing trace functions

Existence

For f an entire function on C, the residue formula gives a integral formula for the trace of the function f computed from the symmetric functions σ of x 1 , . . . , x k :

T (f )(σ) = 1 2iπ . |ζ|=R f (ζ). P σ (ζ).dζ P σ (ζ) = Pσ(x j )=0 f (x j ) ( 7
)
where R is large enough compare to σ.

But for computing the second order partial derivatives of T (f ) this formula is not so convenient.

Remark that for R large enough the quotient P σ (ζ) ζ k is near enough to 1 on the circle |ζ| = R, so we may integrate by part the previous formula and obtain a better formula for the computation of partial derivatives in σ.

Lemma 4.1.1 For any entire function f and any σ ∈ C k we have, for R large enough

T (f )(σ) = - 1 2iπ . |ζ|=R f (ζ).Log P σ (ζ) ζ k .dζ + k.f (0) (8)
Proof. Thanks to the remark above, if we fix σ in a compact subset of N , we may choose R large enough to dispose of the holomorphic 1-form (LogP σ (ζ) ζ k ).dζ around the circle {|ζ| = R} and to integrate by part the formula (7). We have for z near this circle :

d dz f (z).Log P σ (z) z k = f (z).Log P σ (z) z k + f (z).P σ (z) P σ (z) -k.f (z) z
and this implies the formula (8).

Now we may use this formula to compute the second order partial derivatives of T (f ).

Proposition 4.1.2 For any entire function f on C the function T (f ) satisfies the following partial differential system, where we note

∂ h := ∂ ∂σ h : ∂ 2 ∂ p ∂ q - ∂ 2 ∂ p+i ∂ q-i [T (f )] = 0 ∀p, q, i such that p, q, p + i, q -i ∈ [1, k] (@) T m [T (f )] = 0 with T m := ∂ 1 .∂ m-1 + k h=1 σ h .∂ h .∂ m + ∂ m , ∀m ∈ [2, k] Proof. (∂ p [T (f )](σ) = (-1) p-1 . 1 2iπ . |ζ|=R f (ζ). ζ k-p .dζ P σ (ζ) (9) (∂ q ∂ p [T (f )](σ) = (-1) p+q . 1 2iπ . |ζ|=R f (ζ). ζ 2k-q-p .dζ P σ (ζ) 2 (10)
and this depends only of the sum p + q, so this proves the first equations in (@).

In the sequel we shall use the notation:

A(p, q, i) := ∂ 2 ∂ p ∂ q - ∂ 2 ∂ p+i ∂ q-i when p, q, p + i, q -i are in [1, k].
Let now µ be any integer between [0, k -2] and write

ζ µ .P σ (ζ) = k 0 (-1) h .σ h .ζ k-h+µ with σ 0 := 1.
Multiply by f (ζ), divide by P σ (ζ) 2 and integrate on |ζ| = R leads to

∂ k-µ [T (f )](σ) = k-1 h=0 σ h .∂ k-µ-1 ∂ h+1 + σ k .∂ k-µ .∂ k [T (f )](σ)
so the differential operator

T µ 0 := k-1 h=0 σ h .∂ k-µ-1 ∂ h+1 + σ k .∂ k-µ .∂ k + ∂ k-µ (11)
kills the function T (f ) for any choice of the entire holomorphic function f :

C → C. Now, for h ∈ [1, k -1] we have also that ∂ k-µ-1 ∂ h+1 -∂ k-µ .∂ h also kills T (f ) (this differential operator is equal to -A h,k-µ,1
). So we obtain, with m := k -µ

T m = T k-m 0 - k-1 h=1 σ h .A h,m,1 (12)
also kills T (f ) for any f , proving the last equations in (@).

Remarks.

1. All operators in the system (@) have order 2.

2. We have, for any h ∈ [1, k]:

[∂ h , T m ] = ∂ m .∂ h (13)
3. In (@) we may assume that i takes only the value 1 because we have the relations A p,q,i+1 = A p,q,i + A p+i,q-i,1

when p, q, p + i + 1, q -i -1 are in [1, k].

4. Using the formula

∂x j ∂σ h = (-1) h-1 . x k-h j P σ (x j ) ∀h ∈ [1, k]
where x j is a local branch of root of P σ near a point σ 0 ∈ N \ {∆(σ) = 0} one can check directly that the function x q j is solution of the system (@) for any q ∈ N. But this computation is quite involved.

Symbols

We shall study now the ideal in C[σ 1 , . . . , σ k, η 1 , . . . , η k ] (the algebra of polynomial functions on T * N ) generated by the symbols of the differential operators in (@) (note that i = 1 is enough from the remark 3. above).

Lemma 4.2.1 Let l σ (η) := k h=1 σ h .η h . Then the symbol of T m is given by

Symb(T m )(σ, η) = η 1 .η m-1 + l σ (η).η m . (15)
and the ideal in C[σ, η] generated by the symbols of the operators A p,q,1 and T m is equal to the ideal generated by the (2, 2) minors of the matrix

    η 1 -l σ (η) η 2 η 1 . . . . . . η k η k-1     (16)
Proof. It is obvious as the symbol of T m is equal to η 1 .η m-1 + l σ (η).η m .

One main ingredient in our proof of the theorem 5.1.1 is the following proposition. Proof. The first step is to show that this ideal is reduced. This will be given by the proposition 4.2.6. The second step will be given by the lemma 4.2.4.

Denote by I Z the ideal generated by the (2, 2)-minors m α , α ∈ A of the (k, 2)-matrix in ( 16)

Lemma 4.2.3 Each η i .η j for (i, j) in [1, k] × [1, k] may be written as

η i .η j = α u i,j,α .m α + η k .v i,j (17)
where u i,j,α are in C[σ 1 , . . . , σ k ] (so independent of η) and v i,j are polynomials in (σ, η) linear in η (so at most of degree 1 in η 1 , . . . , η k ).

Proof. We shall make a descending induction on h ∈ [1, k] to prove the following assertion:

• For a given h ∈ [1, k] we can write η j .η h as in ( 17), for each j ∈

[1, k].
As this is clearly true for h = k, let us assume that for some h ∈ [1, k -1] we have proved the assertion for h + 1 and we shall prove it for h. As η 1 .η h + k p=1 σ p .η p .η h+1 is in I Z the induction hypothesis gives that η 1 .η h may be written as in (17). For j ∈ [2, k] we have η j .η h -η j-1 .η h+1 ∈ I Z and then the induction hypothesis allows to completes the proof of our induction step. Proof. Consider the polynomial map Φ :

C k × C 2 → C k × C k given by Φ(s, ζ 0 , ζ 1 ) = (σ, η) with σ h := (-1) h .s h and η h := ζ h 0 .ζ k-h 1 ∀h ∈ [1, k].
Define also the hypersurface

H in C k × C 2 H := {(s, ζ 0 , ζ 1 ) ∈ C k × C 2 / k h=0 (-1) h .s h .ζ h 0 .ζ k-h 1 = 0} with the convention s 0 ≡ 0. Remark that H := H ∩ {ζ 0 = 1} is a smooth connected k-dimensional sub-manifold in H (isomorphic to C k via the map (s, ζ 1 ) → (s 1 , . . . , s k-1 , ζ 1 )).
Then we shall first verify that Φ(H) ⊂ |Z|. As the equation of H shows that on the image of H we have In order to prove this we need the following remark.

l σ (η) = -ζ k 1 the vectors     -l σ (η) η 1 . . . η k-1     = ζ 1 .     ζ k-1 1 ζ 0 .ζ k-2 1 . . . ζ k-1 0     and     η 1 η 2 . . . η k     = ζ 0 .     ζ k-1 1 ζ 0 .ζ k-2 1 . . . ζ k-1 0    

Remark. The open set {η

k = 0} ∩ |Z| is the complement of C k ×{0} in |Z|.
This means that on |Z| the equality η k = 0 implies that η = 0 : looking of the (k, 2) matrix whose (2, 2) minors define the ideal I Z we immediately see that η k = 0 implies η k-1 = 0 and then

η k-2 = • • • = η 1 = 0. Of course the subset C k ×{0} is in |Z| because each equation of |Z| is homogeneous of degree 2 in η. Now on {η k = 1} ∩ |Z| we see that the map {η k = 1} ∩ |Z| → H given by (σ, η) → (s, η k-1
) is the inverse of the restriction of Φ to H × {ζ 0 = 1}; so it is an isomorphism. From this, it is easy to deduce, using homogeneity, that the restriction of Φ to So fix a point (σ, 0) ∈ C k ×{0} and assume that σ = 03 . Let z be a non zero root of the polynomial k h=0 σ h .z k-h = 0 and let ε > 0. Then, as the point (s, ε, ε.z) is in H ∩ {ζ 0 = 0}, the point Φ(s, ε, ε.z) is in |Z| ∩ {η k = 0} and when ε → 0 it converges to (σ, 0) concluding the proof.

H ∩ {ζ 0 = 0} → |Z| ∩ {η k = 0} is a k-sheeted etale cover corresponding to the k-th root of η k . As H ∩ {ζ 0 = 0} is isomorphic to H × C * which is smooth connected of dimension k + 1,
Note that the previous lemma implies that if a holomorphic function f on N × C k vanishes on |Z|, then Φ * (f ) vanishes on H. Lemma 4.2.5 Let f be an holomorphic function on N × C k which is homogeneous of degree 0 or 1 in η and which vanishes on |Z|. Then f ≡ 0.

Proof. In the case where f is independent of η, then Φ * (f ) is independant of ζ 0 and ζ 1 and vanishes on H. So Φ * (f ) vanishes. But Φ induces an isomorphism of C k ×{0} to C k ×{0} and then f ≡ 0. In the case of a function linear in η we may write f

= k h=1 f h (σ).η h and then Φ * (f ) = k h=1 g h (s).ζ h 0 .ζ k-h 1 . As Φ * (f ) vanishes on H we obtain 4 that g h ≡ 0 for each h ∈ [1, k] and then f ≡ 0. Proposition 4.2.6 Let f be a holomorphic function on N × C k which vanishes on |Z|. Then f is in I Z .
Proof. It is enough to consider the case where f is homogeneous in η because the general case follows immediatly: Consider the partial Taylor expansion f (σ, η) = ∞ d=0 f d (σ, η) where f d is homogeneous of degree d in η. Then the homogeneity of |Z| implies that f vanishes on |Z| if and only if each f d vanishes on |Z|. Now, as the cases d = 0, 1 are already obtained in lemma 4.2.5, consider the set of holomorphic functions on N × C k which are homogeneous of some degree d ≥ 2 in η, vanish on |Z| and are not in I Z . Assume that this set is not empty (if this is not the case, we are done, thanks to the lemma 4.2.5). Assume that f has minimal degree d 0 ≥ 2 in this set and write

f (σ, η) = η k .g(σ, η) + h(σ, η )
where η := (η 1 , . . . , η k-1 ). Here h is independent of η k homogeneous of degree d 0 ≥ 2 in η and g is homogeneous of degree d 0 -1 in η. The function h is in the ideal generated by η i .η j , (i,

j) ∈ [1, k -1] 2 in O(C k )[η ]
and, thanks to the lemma 4.2.3 we may write

h(σ, η ) = α h α .m α + η k .v(σ, η)
where m α are the generators of I Z (which are homogeneous of degree 2 in η), where the functions h α are holomorphic functions (independent of η k ) homogeneous of degree d 0 -2 in η and where v(σ, η) is homogeneous in

η of degree d 0 -1. So we obtain f (σ, η) = η k .(g -v)(σ, η) + α h α .m α .
Now the function f := fα h α .m α is again vanishing on |Z| (because the functions m α vanish on |Z|) and is homogeneous of degree d 0 . As we have f (σ, η) = η k .g(σ, η) where g := g -v is homogeneous of degree d 0 -1 the irreducibility of |Z| implies either η k = 0 on |Z| or g = 0 on |Z|. But we have seen in the lemma 4.2.4 that η k does not vanish at the generic point in |Z|. So we find a homogeneous degree d 0 -1 function g vanishing on |Z| and the minimality of the degree d 0 of f implies :

• either d 0 -1 ≤ 1 and then g ≡ 0 by the lemma 4.2.5 which implies f ≡ 0 and so f is in I Z . Contradiction.

• or d 0 -1 ≥ 2 and then g has to be in I Z because of the minimality of d 0 and again f is in I Z . Contradiction.

So every holomorphic function homogeneous in η vanishing on |Z| is in I Z .

Remark. It is an easy corollary of the previous proposition that the sheaf I Z of ideals in O N ×C k is reduced.

In the sequel we simply note Z the common zero subset in N × C k T * N of the minors (2, 2) of the matrix in ( 16), which is the characteristic cycle (which is reduced) of the system (@).

The simple corollary of the proposition 4.2.2 which follows will be used later on.

Corollary 4.2.7 The Zariski open set Z ∩ {∆(σ).η 1 = 0} in Z is dense in Z.
Proof. In fact we have seen that the set {η 1 = 0} ∩ Z is the union of N × {0} and {σ k = 0} ∩ Z. But Z has pure dimension k + 1 and has a dimension 1 fiber over each point in N . The analytic subsets N × {0}, {σ k = 0} ∩ Z and {∆(σ) = 0} ∩ Z have pure dimension k so have no interior point in Z. Proof. It is enough to prove the proposition when U is the image by s of the product

U 0 := D 1 × . . . . . . D k of k disjoint discs in C.
In this case Ξ induces an isomorphism on U and Q = Ξ(P ) where P is a S k -invariant differential operator on U 1 := s -1 (U). Then we reduce the problem to show the following two facts :

i) If P kills any Newton function N m (x), m ∈ N on U 1 then the symbol of P vanishes on Y ∩ (U 1 × C k ) (T * M ) |U 1 where Y := {(x, ξ 1 , . . . , ξ k ) ∈ M × C k / ξ i .ξ j = 0 ∀i = j}.
ii) The map SΞ * : s -1 (T * N ) → T * M sends s -1 (Z) into Y , and this induces a bijection between these sub-sets over the dense open set s -1 {∆(σ) = 0} in M .

The proof of i) will use the following two lemmas and the corollary 4.2.11. Lemma 4.2.9 Let J 0 be the left ideal in D M generated by the differential operators

∂ 2 ∂x i ∂x j for (i, j) ∈ [1, k] 2 , i = j,
and let J 1 be the left ideal in s * (D M ) S k which is given by its intersection with s * (J 0 ). Let U 1 ⊂ M be an open set as above and let P be a S k -invariant differential operator on U 1 . Then P may be written

P = N q=0 k-1 p=0 s * (a p,q ).x p j .( ∂ ∂x j ) q modulo Γ(U, J 1 )
where a p,q are holomorphic functions on U := s(U 1 ).

Proof. Let c.x α .( ∂ ∂x ) β be a monomial of P with c = 0. If β = (0, . . . , 0, q, 0, . . . , 0) then the symmetrization P 1 of this monomial is in J 1 and P -P 1 neither have this monomial nor these monomials deduced from it by the action of S k . So we may write

P = α N q=0 c α,j .x α .( ∂ ∂x j ) q modulo J 1 .
Now write x α = (x(j)) α(j) .x p j , where we use the notation : x(j) := (x 1 , . . . , xj , . . . , x k ) and α(j) = (α 1 , . . . , αj , . . . , α k ). The S k -invariance of P implies the invariance of c α,j .(x(j)) α(j) by the stabilizer S k (j) of j. Using the lemma 3.2.4, we obtain that P has the following form

Q = k-1 p=0 N q=0 s * (a p,q ).x p j .( ∂ ∂x j ) q
concluding the proof.

Lemma 4.2.10 Let x 1 , . . . , x k be distinct points in C and fix an integer N . Let P q be the C -vector space of polynomial of degree at most q in C[z]. Then the linear map

L : P k.(N +1)-1 → C k.(N +1) , Q → Q (p) (x j ), ∀j ∈ [1, k], ∀p ∈ [0, N ] is bijective.
Proof. This is, of course, a degenerate case of the standard Lagrange interpolation. We shall give a quick proof of it because the reader may be not so familiar with this very degenerate case of Lagrange interpolation. Consider the ideal I in O C of germs vanishing at order N at each point +1) analogous to L. Now each entire function f on C may be written, thanks to Weierstrass division theorem5 , in an unique way as

x j for j ∈ [1, k]. Then the exact sequence of coherent sheaves 0 → I → O → O I → 0 gives a surjective linear map L 1 : O(C) → C k.(N
f (z) = g(z).P (z) N +1 + q(z)
where g is an entire function, P (z) := k j=1 (z -x j ) and q is a polynomial of degree at most k.(N + 1) -1. Then this allows to conclude that L is surjective as

L 1 (f ) = L(q).
Corollary 4.2.11 Let P be a S k -invariant differential operator on U 1 which can be written as follows :

P := N q=0 k-1 p=0 s * (a p,q ).x p j .( ∂ ∂x j ) q
where a p,q are holomorphic functions on U = s(U 1 ). Assume that P kills any Newton function N m (x), m ∈ N on U 1 . Then P = 0.

Proof. It is enough to prove that each a p,q vanishes at each point in U ∩{σ k = 0}. Assume that there exists p 0 , q 0 and σ 0 = s(x 0 1 , . . . , x 0 k ) ∈ U satisfying σ 0 k = 0 and such that a p 0 ,q 0 (σ 0 ) = 0. Then we may find, thanks to the lemma 4.2.10, a polynomial Π of degree at most k.(N + 1) -1 such that k j=1 Π (q 0 ) (x 0 j ).(x 0 j ) p 0 = 0 and such that all derivatives at each point x 0 1 , . . . , x 0 k of order ≤ N and not equal to q 0 (including 0 when q 0 = 0) vanish. This is possible because for each j ∈ [1, k] we have x 0 j = 0. Then we obtain a contradiction because P has to kill the trace function T (Π)(σ) := k j=1 Π(x j ) as P has to kill each Newton function N m (x) and T (Π) is a finite linear combination of these functions. But our choice of Π implies P [T (Π)] = 0 at σ 0 . Contradiction. So P = 0.

Proof of i) in 4.2.8. The three results above prove more that the fact i) stated above because we obtain that P is a section on U of the left ideal J 1 of s * (D M ) S k . And the lemma 2.2.7 shows that this ideal is globally generated by S 2 , . . . , S k the elementary symmetric functions of the differential operators ∂ ∂x j , j ∈ [1, k].

The proof of ii) in 4.2.8 will use the following two lemma.

Lemma 4.2.12 Let Z be the subspace of C k × C k defined by the ideal generated by the (2, 2) minors of the matrix (16). If (σ, η) belongs to Z and satisfies η 1 = 0, then we have :

1. l σ (η) = 0; 2. for each h ∈ [1, k], η h = η 1 .(-η 1 /l σ (η)) h-1 ;
3. l σ (η)/η 1 is a (non zero) root of the polynomial P σ .

Proof. The vanishing of the first minor equal to η 2 1 + l σ (η).η 2 implies that l σ (η) does not vanish. So 1. is proved. Assume that we have proved that 2. is valid for h ∈ [1, p] for p ∈ [1, k -1]. Note that this is clear for h = 1. Then the minor η 1 .η p + η p+1 .l σ (η) vanishes on Z and gives

η p+1 = -η 1 .η p /l σ (η) = η 1 . -η 1 /l σ (η)
p using the induction hypothesis. So 2. is proved. Now write l σ (η) = k h=1 σ h .η h and replace η h by the formula 2. This gives, after multiplication by l σ (η) k-1 and dividing by η k 1 (l σ (η)/η 1 ) k = k h=1 (-1) h-1 .σ h . l σ (η)/η 1 k-h so P σ (l σ (η)/η 1 ) = 0 proving 3.. In our next lemma we shall use now the notations introduced in the beginning of the paragraph 3.2.

Lemma 4.2.13 Fix σ ∈ C k and consider η ∈ C k given by η h = a h-1 η 1 , h ∈ [1, k]
where a ∈ C * is given. Then we have for a.z = -1:

k h=1 Θ h (z, σ).η h = (-a) k 1 + a.z . P σ (z) -P σ (-1/a) .η 1
and for a.z = -1:

k h=1 Θ h (z, σ).η h = z -k .P σ (z).η 1
Proof. Note first that, by homogeneity of degree 1 in η we may assume that η 1 = 1. As, by definition we have

k h=1 Θ h (z, σ).η h = k h=1 h-1 p=0 (-z) h-p-1 .σ p .η h
our hypothesis gives for a.z = -1:

k h=1 Θ h (z, σ).η h = k h=1 (-a.z) h-1 h-1 p=0 σ p .(-z) -p k h=1 Θ h (z, σ).η h = k-1 p=0 σ p .(-z) -p k h=p+1 (-a.z) h-1 .η 1 k h=1 Θ h (z, σ).η h = k-1 p=0 σ p .(-z) -p (-a.z) p (-a.z) k-p -1 -(1 + a.z)
and then:

k h=1 Θ h (z, σ).η h = - (-a) k 1 + a.z . k-1 p=0 (-1) p .σ p .z k-p + 1 1 + a.z k-1 p=0 σ p .a p but as (-a) k .P σ (-1/a) = (-a) k . k p=0 (-1) p .σ p .(-1/a) k-p = k p=0 σ p .a p we obtain k h=1 Θ h (z, σ).η h = - (-a) k 1 + a.z P σ (z) -(-1) k σ k + (-a) k 1 + a.z P σ (-1/a) -(-1) k .σ k k h=1 Θ h (z, σ).η h = - (-a) k 1 + a.z P σ (z) -P σ (-1/a)
For a.z = -1 the computation gives, again for η 1 = 1:

k h=1 Θ h (z, σ).η h = k h=1 (-a.z) h-1 h-1 p=0 σ p .(-z) -p k h=1 Θ h (z, σ).η h = k-1 p=0 (k -p).σ p (-z) -p = z -k+1 .P σ (z)
concluding the proof.

Remarks.

1. When z and -1/a are distinct roots of P σ we find k h=1 Θ h (z, σ).η h = 0. If z = -1/a is a double root of P σ we find again k h=1 Θ h (z, σ).η h = 0.

2. Assume now that η 1 = • • • = η k-1 = 0 and z = 0. Then we obtain Θ(z, σ, η) = (-1) k-1 z . P σ (z) -(-1) k σ k .η k so we find again k h=1 Θ h (z, σ).η h = 0 when σ k = 0 and P σ (z) = 0.

3. At any point (x, s(x), η) in s -1 Z \ (N × {0}) with ∆(s(x)) = 0 we have at least (k -1)-distinct roots x i , i ∈ [1, k], of the polynomial P s(x) such that the numbers k h=1 Θ h (x i , s(x)).η h vanishes. So the symmetric functions of order ≥ 2 of the numbers k h=1 Θ h (x i , s(x)).η h , i ∈ [1, k] vanish at such a point.

Proof of ii) of 4.2.8. As Y is the union of the sub-manifolds

Y j := {(x, α 1 , . . . , α k ) ∈ T * M / α 1 = • • • = αj = • • • = α k = 0} for j ∈ [1, k],
it is enough to show that for any point (x, s(x), β 1 , . . . , β k ) in s -1 (Z) we have T * s (x, s(x), β 1 , . . . , β k ), e j = 0 (18

) for k -1 values of j ∈ [1, k].
In fact it is enough to prove this fact assuming that β 1 = 0 and ∆(s(x)) = 0, because these two conditions define an open dense subset in s -1 (Z) thanks to the corollary 4.2.7.

Then we obtain from lemma 4.2.12 that there exists j 0 ∈ [1, k] such that we have

β h = β 1 (-1/x j 0 ) h-1 , ∀h ∈ [1, k] with P s(x) (x j 0 ) = 0.
Then the lemma 4.2.13 with the remark 3. which follows it, gives (18) for each j = j 0 concluding the proof of the proposition 4.2.8.

The solution

Notations. Let J be the left ideal of D N generated by the elements in the system (@). Let U 0 := k h=1 h.σ h .∂ h and for q ∈ N let K q the left ideal in D N generated by the differential operators in (@) and U 0 -q. So K q = J + D N .(U 0 -q).

The main theorem

Our main result is the following characterization of "trace functions". Proof. Remark first that P kills any local trace function on an open set U in N by the lemma 3.1.4 and so, thanks to the proposition 4.2.8, if P is not 0 the symbol of P vanishes on Z. But using the proposition 4.2.6 we may find P 1 ∈ Γ(N, J ) such that the symbol of P 1 is equal to the symbol of P . Then P -P 1 again kills any Newton polynomial N m , m ∈ N and has order strictly less than the order of P . So if we assume that there exists such a P ∈ Γ(N, D N ) which is not in Γ(N, J ), choosing such a P with minimal order with these properties we obtain a contradiction by the previous argument. This conclude the proof.

Corollary 5.1.2 The D N -module M := D N J is sub-holonomic. For each q ∈ N the D N -module N q := D N K q is holonomic on the complex manifold N . Its solutions of order 0 on any simply connected open set U contained in {∆(σ) = 0} are the linear combinations (with constant coefficients) of the q-th power of the Then the first formula is now consequence of the following computation

[∇, ∂ 1 .∂ h-1 ] = -(k -1).∂ 2 .∂ h-1 -(k -h + 1).∂ 1 .∂ h = -(2k -h).∂ 1 .∂ h modulo (∂ 1 .∂ h -∂ 2 .∂ h-1 )
for each h ∈ [2, k -1] using our previous computations :

[∇, T h ] = [∇, ∂ 1 ∂ h-1 ] + [∇, E∂ h ] + [∇, ∂ h ] = -(2k -h).∂ 1 ∂ h + k.∂ 1 ∂ h -(k -h).∂ h+1 modulo A 1,h,1 = -(k -h).T h+1 modulo A 1,h,1
and we find [∇, T k ] = 0 for h = k. Remark that, using the notation of the first sections, we have ∇ = Ξ(S 1 ) = Σ 1 because these derivations coincide on the elementary symmetric functions σ 1 , . . . , σ k .

A holonomic system for trace forms

DN m (σ) = 1 2iπ |ζ|=R ζ m+k-1 .dζ P σ (ζ) . ( 20
) v) For each h ∈ [1, k] and each m ≥ 0 we have ∂N m ∂σ h = (-1) h-1 .m.DN m-h . (21)
Proof. Remark first that the formula (20) define a holomorphic function on N because when σ stays in a fixed relatively compact set the integral is independent of the choice of R large enough. The residue formula implies easily that for ∆(σ) = 0 this holomorphic function satisfies (19). Now, assuming that we have

|σ h | ≤ ε.R for each h ∈ [1, k] with ε 1/k we obtain the estimate |DN m (σ)| ≤ R m 1 + k.ε .
This implies that DN m is a polynomial of degree at most equal to m. The pure weight m of this polynomial is easily obtained by a change of variable in (20).This gives i) and the fact that

DN m is in C[σ 1 , . . . , σ k ] for each m ∈ N.
With our definition of the polynomials DN m the formula ii) is obvious. We show the first assertion in iii) by induction on m ≥ 0 : For m = 0 the relation

ζ k = P σ (ζ) + k h=1 (-1) h-1 .σ h .ζ k-h
and the relation ii) gives DN 0 ≡ 1. Then the induction step is given by the relation ii)

as σ 0 = 1. So DN m is in Z[σ 1 , . . . , σ k ] for each m ≥ 0. To complete the proof of iii) it is enough to recall that for h ∈ [1, k] the polynomial DN h is in Z[σ 1 , . . . , σ h ],
has pure weight h and that the coefficient of σ h in it is equal to (-1) h-1 . The formula in v) is easily obtained by derivation of the formula (8) for f (z) = z m using iv). Theorem 5.2.3 Assume that the function G ∈ Γ(N, O N ) is a trace form; then it satisfies the system ( @) which is generated by the differential operators A p,q for p, q, p + 1, q -1 in Proof. First remark that if G is the trace form of the entire function g = m≥0 γ m .z m , we have G = m≥0 γ m .DN m-k+1 with uniform convergence on compact set. Thanks to formula (21) the two assertions are easy consequences of the theorem 5.1.1 using the following commutation relations [A p,q , ∂ h ] = 0 ∀p, q, p + 1, q -1 ∈

[1, k], ∀h ∈ [1, k] (T m + ∂ m ).∂ h = ∂ h .T m ∀m ∈ [2, k], ∀h ∈ [1, k]
because they show that when F is a solution of (@) then ∂ h F is solution of ( @) for each h ∈ [1, k].

Remarks.

1. As an easy consequence of the previous theorem, using the fact that the characteristic variety of the left ideal J in D N generated by ( @) is again equal to Z which is irreducible, we obtain that the left ideal J in D N is the annihilator of all trace forms, or equivalently, of all polynomials DN m , m ∈ N.

2. Any component of a global solution of the order 1 linear system satisfied by the Lisbon integrals (see [B.MF 19]) is a trace form and then satisfies the system ( @). Note that the components of such a solution are, up to the order and some signs, the derivatives ∂ 1 F, . . . , ∂ k F of a trace function F . So global solutions of the system described in loc. cit. may be identified with differential of trace functions.

6 Miscellanous complements Proof. First remark that each U d for d ≥ k is in the C[x 1 , . . . , x k ] S k -module generated by U 0 , . . . , U k-1 because, by definition we have for each integer q the relation k h=0 (-1) h .σ h .x q+k-h j = 0 for each j ∈ [1, k], where σ h ∈ C[x 1 , . . . , x k ] S k is the h-th symmetric function of x 1 , . . . , x k . Let now δ := k j=1 a j (x). ∂ ∂x j be a symmetric derivation. Writing for each j ∈ [1, k] a j (x) := N p=0 a j,p (x(j)).x p j , where x(j) := (x 1 , . . . , xj , . . . , x k ), we see that the S k -invariance of δ implies that for each p the polynomial a j,p is S k [j]-invariant, where S k [j] is the stabilizer of j in S k . Then, if σ h [j] denote the h-th symmetric function of x(j), using the fact (see lemma 3.2.4) that σ h (j) = h q=0 σ h-q .(-x j ) q for each h ∈ [1, k -1] (with the convention σ 0 := 1), we may write is an element of order 1 in W 2 with pure weight at least equal to -1. As Ξ induces an isomorphism on elements of order 0, this implies that if any derivation in W 2 which has pure weight q is in the image of Γ(N, Ξ) : W S k 1 → W 2 , then q ≥ -1. This shows that ∂ h for h ∈ [2, k] are not in this image. So Γ(N, Ξ) is not surjective (and Ξ also is not surjective). Lemma 6.1.2 For each non negative integer p define for x 1 , . . . , x k with elementary symmetric functions σ 1 , . . . , σ k such that ∆(σ) = 0 (-1) h-1 x k-h+m-1 j P σ (x j ) = (-1) h-1 .m.DN m-h

  The natural action of S k on M = C k defines an action on M ×N (the action on N is, of course, trivial) and also on G. For this action p 1 and p 2 are equivariant. Then we have an action of S k on the sheaves (p 2 ) * (p * 1 (D M )) and (p 2 ) * (p * 2 (D N )). We shall denote respectively by (p 2 ) * (p * 1 (D M )) S k and (p 2 ) * (p * 2 (D N )) S k the subsheaves of S k -invariant sections of these sheaves. They are O N -sub-modules of these O N -algebras. • We have also an action of S k on the sheaf of O N -algebras s * (D M ) and we shall note s * (D M ) S k the corresponding O N -sub-modules of the sheaves of S k -invariant sections. • As s is the (holomorphic) quotient map (see [B.M 1] chapter I theorem 3.1.13) we have a natural isomorphism θ : s * (O M ) S k → O N whose inverse is simply the pull-back of holomorphic functions. Theorem 3.2.1 There exists a natural morphism of sheaves of O N -algebras Ξ : s * (D M ) S k → D N such that, for any P ∈ s * (D M ) S k and for any f ∈ s * (O M ) S k we have θ(P [f ]) = Ξ(P )[θ[f ]] (5) Moreover, this morphism is injective and it restriction to the Zariski open set {σ ∈ N / ∆(σ) = 0} in N is an isomorphism of sheaves of O N -algebras.

  To prove the formula (5) it is enough to prove it on the dense Zariski open set {σ ∈ N / ∆(σ) = 0} in N . So consider D 1 , . . . , D k disjoints open discs in C and let U 0 := D 1 × • • • × D k and U := s(U 0 ). Then s induces an isomorphism of U 0 onto U. If U 1 := s -1 (U), a S k -invariant function (resp. S k -invariant differential operator) on U 1 is simply given by a function (resp. a differential operator) on U 0 or equivalently by a function (resp. by a differential operator) in U via the isomorphism s |U 0 . Then Ξ |U is an isomorphism of s * (D M ) S k |U s * ((D M ) |U 0 ) (D N ) |U and the formula (5) is satisfied on U as θ reduces to the composition of the restriction to U 0 with s. Corollary 3.2.3 In the situation of the previous theorem the map Ξ induces a graded sheaf map between commutative graded O N -algebras :

  The ideal in C[σ 1 , . . . , σ k, η 1 , . . . , η k ] generated by the (2, 2)-minors of the matrix in (16) is prime.

  Lemma 4.2.4 Let |Z| the algebraic subset of common zeros of the polynomials in the ideal I Z . Its Zariski open set {η k = 0} is dense, smooth and connected of dimension k + 1. So |Z| is irreducible.

  are proportional; this allows to conclude that Φ(H) ⊂ |Z|. Now we have {ζ 0 = 0} ∩ H H × C * by the map (s, ζ 0 , ζ 1 ) → ((s, ζ 1 /ζ 0 ), ζ 0 ) and H is smooth of dimension k and connected. But on the open set {ζ 0 = 0} ∩ H the restriction of the map Φ is an etale k-sheeted covering onto {η k = 0} ∩ |Z|:

  the only missing point is to prove the density of the open set {η k = 0} ∩ |Z| in |Z|.

  Proposition 4.2.8 Let U be a non empty connected open set in N and Q a non zero section on U of D N which annihilates any Newton polynomial N m (σ), m ∈ N on U. Then the symbol of Q vanishes on Z |U ⊂ (T * N ) |U .

  Theorem 5.1.1 Let P ∈ Γ(N, D N ) be a differential operator which kills any Newton polynomial N m , m ∈ N. Then P is in the left ideal Γ(N, J ) of Γ(N, D N )

  So we have a right action of ∇ on the D N -module M = D N J . Note that [∇, U 0 ] = ∇ as ∇ is a derivation of weight -1. So the action of ∇ on M shifts the weight decomposition of the global sections of M by -1. In the bijective correspondence between global 0-solutions 6 of M and Γ(C, O C )-given by f → T (f ) it gives the usual derivation of entire functions.

For

  each m ∈ Z, m ≥ -k + 1 and for each σ ∈ N such that ∆(σ) Each DN m is the restriction to the open set {∆(σ) = 0} of a polynomial of (pure) weight m in C[σ 1 , . . . , σ k ] and the following properties are satisfied : i) For m ∈ [-k + 1, -1], DN m = 0. ii) For each m ≥ 1, k h=0 (-1) h .σ h .DN m-h = 0 with the convention σ 0 ≡ 1. iii) The polynomials DN m are in Z[σ 1 , . . . , σ k ] and DN 1 , . . . , DN k generate this Z-algebra. iv) For R ||σ|| we have for each m ≥ -k + 1:

  Definition 5.2.2 We shall call DN m the m-th derived Newton polynomial in x 1 , . . . , x k . We shall say that G ∈ Γ(N, O N ) is a trace form if there exists g ∈ Γ(C, O C ) Let H := {(σ, z) ∈ N ×C / P σ (z) = 0} and π : H → N the projection. For g ∈ Γ(C, O C ) we have the equalityT race π (g(z).dz ∧ dσ 1 ∧ • • • ∧ dσ k-1 ) = T (g)(σ).dσ 1 ∧ • • • ∧ dσ kwhich explains our terminology "trace form". This also prove the holomorphy of G for any g ∈ Γ(C, O C ).

  [1, k] and byT m := T m + ∂ m for m ∈ [2, k]. Moreover, for any h ∈ [1, k] there exists a trace function F h such that G = ∂ h F h .

6. 1

 1 Symmetric derivations Lemma 6.1.1 The C[x 1 , . . . , x k ] S k -module of the symmetric derivations in W S k 1 is generated by U 0 , . . . , U k-1 where we define for each d ∈ N

  σ 1 , . . . , σ k ).U p because the S k -invariance of δ implies that b j,p does not depend on j ∈ [1, k] for each p.Remark. As the derivation U d has pure weight d -1 this implies that the image by Ξ of any pure weight symmetric derivation in W S k 1

  any Q ∈ C[x 1 , . . . , x k ] S k , ∇ p (Q) is in C[x 1 , . . . , x k ] S k . So ∇ p define a derivation in W 2 with pure weight equal to p -k. For each p in [0, k -1] we have ∇ k-p = (-1) p-1 . ∂ ∂σ p(24)Proof. The first assertion is an easy consequence of proposition 5.2.1 as we have∇ h [N m ] = h-1 .m.DN m-h ∀p ∈ [1, k] ∀m ∈ N(25) To prove (24), as a derivation in W 2 which vanishes on the Newton polynomials N 1 , . . . , N k is nul, it is enough to compare the values of ∇ k-p and ∂ p on N m for m ∈ [1, k]. But for any m ∈ N we have:

  x k ). Let F = ν≥0 P ν be its Taylor expansion at the origin in C k . Each homogeneous polynomial P ν is symmetric and as the differential operators S h are homogeneous we obtain that S h (P ν ) ≡ 0 for each ν ≥ 0 and each h ∈ [2, k] by uniqueness of the Taylor expansion. Then we can find polynomials R ν ∈ C[x 1 ] which are homogeneous of degree ν such that σ 1,k [R ν ] = P ν . From the convergence of the Taylor series of F , it is then easy to see that the series ν≥0 R ν converges uniformly on compact sets in C and defines an entire function f such that T (f ) = F . As the converse is obvious, the proposition is proved. The intersection of the ideal I in C[ξ 1 , . . . , ξ k ], generated by ξ i .ξ j for i = j with C[ξ 1 , . . . , ξ k ] S k in the ideal generated by the elementary symmetric functions S 2 , . . . , S k of ξ 1 , . . . , ξ k .Proof. Remark first that S 2 , . . . , S k are in I and symmetric. Let P ∈ C[ξ 1 , . . . , ξ k ] S k be in I. Then as P is in C[S 1 , . . . , S k ] we may write

	Consider now a symmetric entire function F such that S h (F ) = 0 for each h ∈ [2, k].
	Proposition 2.2.6 Any element in W S k 1	which kills any Newton polynomial
		k
		N m (x) :=	x m j , m ∈ N
		j=1
	is in the left ideal of W S k 1	generated by S 2 , . . . , S k .
	The proof will be a simple consequence of the following lemma.
	Lemma 2.2.7

S h-1 (σ h,k (Q)) = S h-1 (σ h-1,k (Q)) = (h -1)! 2 .σ h-1,k ∂ h-1 R ∂x 1 . . . ∂x h-1

ε * 1 , . . . , ε * k denotes the dual basis of ε 1 , . . . , ε k .

The reader may consult[Bj] or[Bor] for basic results on D-modules.

This will be enough to prove our density statement !

for instance put ζ 0 = 1 and use the fact that 1, z, . . . , z k-1 is a basis of O H over O C k .

see [B-M 1] ch.II th. 3.2.9.

so Γ(N, Hom D N (M, O N )).

branches on U of the multivalued function z on N which is the solution of the equation

Proof. As any q-th power of a local branch of the multivalued function z is a local trace function, the differential operators in (@) kills such a local branch. But any q-th power of such a local branch is also killed by U 0 -q. To see that N q is holonomic it is enough to see that Z ∩ { k h=1 h.σ h .η h = 0} has dimension k. But this set is the union of N × {0} with the co-normal to the hypersurface

Remark. It is easy to check the following commutation relations

and this shows that J .U 0 is contained in J . So we have an action of U 0 on the D N -module M giving a spectral decomposition

where F m is the vector space of differential operators of pure weight m modulo those in Γ(N, J ). Note that P ∈ W 2 has pure weight p if and only if [P, U 0 ] = -p.P . For instance the commutation relations above just mean that A p,q has pure weight -(p + q) and that T m has pure weight -m.

Lemma 5.1.3 Let ∇ := k-1 h=0 (k -h).σ h .∂ h+1 . Then we have the following commutation relations

for all p, q such that p, q, p + 1, q -1, p + 2, q -1 are in [1, k] Proof. Recall that, by definition, we have

Note that the fact that the derivations ∇ p for p ∈ [0, k -1] commute pair-wise is not obvious from a direct computation. A consequence of the previous lemma is the fact that for any P ∈ W 2 of order q then ∆ 2q .P is in the image by Ξ of W S k 1 .

Primitive Newton polynomials

For m ≥ k + 1 the differential form

with the convention σ 0 = 1. Then we have for m ≥ 0 :

with the following conventions : N m-p = 0 for m < p and N m-p /(m -p) = 1 for m = p.

Proof. For m ≥ k + 1 the formula ( 29) is consequence of ( 21) and the fact that

For m < p the left hand-side is clearly 0. For m = p the only term in P N p which depends on σ p is the term (-1) p-1 .N p p and the coefficient of σ p in N p is equal to 7 (-1) p-1 .p concluding the proof.

7 Take the case z p -1 to compute it ! Example. we have for k ≥ 4, P N 1 = -σ 1 , P N 2 = (1/2)σ 2 1 + σ 2 and

and it is easy to check that ( 29) is valid on these examples.

Final remarks. Using the commutation relations given in the proof of the theorem 5.2.3 we see that the left ideal in Γ(N, D N ) which annihilate all polynomial P N m for m ∈ N * is generated by the differential operators A p,q , p, q, p + 1,

This system is given by differential operators with only order 2 terms and has the same characteristic variety than (@) or ( @). So any σ p , p ∈ [1, k] is a solution of this system.

Note that the polynomials

It is easy to see that for m ≥ k + 1 the polynomial P N m (σ) is the trace (via the map π : H → N ) of the restriction to H of the polynomial Q m,σ [z] which is the primitive (in z) of the polynomial z m-k-1 .P σ (z) vanishing at z = 0.

7 References