PhD Julien C Marie
email: julien.marie@inserm.fr

PhD Edith Bonnelye
email: edith.bonnelye@inserm.fr

Effects of estrogens on osteoimmunology: a role in bone metastasis

Bone loss associated with estrogen deficiency indicates a fundamental role of these hormones in skeletal growth and bone remodeling. In the last decades, growing recent evidence demonstrated that estrogens can also affect the immune compartment of the bone. In this review, we summarize the impacts of estrogens on bone immune cells and their consequences on bone homeostasis, metastasis settlement into the bone and tumor progression. We also addressed the role of an orphan nuclear receptor ERRalpha ("Estrogen-receptor Related Receptor alpha") on macrophages and T lymphocytes, and as an immunomodulator in bone metastases. Hence, this review links estrogens to bone immune cells in osteo-oncology.

Introduction

Estrogens family of steroid hormones composed of estradiol, estrone, estriol and estetrol.

Estradiol (E2) is the most prevalent and the most potent estrogen. Though ovaries are the main producers of estrogens, fat tissues, testes, the adrenal cortex, and the liver also contribute to their production [START_REF] Almeida | Estrogens and Androgens in Skeletal Physiology and Pathophysiology[END_REF]. Estrogens are major players in both skeletal growth, particularly during puberty, and skeletal maintenance, including normal bone mineral density and trabecular bone mass, during adult life. Reduced estrogen levels at menopause for women and later in life for men lead to decreased bone density and microarchitecture deterioration resulting in a high risk of fracture [START_REF] Almeida | Estrogens and Androgens in Skeletal Physiology and Pathophysiology[END_REF].

Estrogens exert their actions through genotropic and non-genotropic signaling pathways. They can bind to two nuclear receptor isoforms, estrogen receptor-a (ERa) and ERb which, once their ligand pocket is occupied, migrate to the nucleus and fulfil their roles as transcription factors by binding to dedicated DNA sequences [START_REF] Evans | Nuclear Receptors, RXR, and the Big Bang[END_REF]. ERa can also activate kinases (MEK, ERK, JNK) and modulates several transcription factor activities (c-jun, Elk-1) (3). More recently, a G Protein-Coupled Estrogen Receptor (GPER1) has been discovered. Unlike ERa/b, GPER1 binds estrogens with lower affinity [START_REF] Zimmerman | GPER-novel membrane oestrogen receptor[END_REF]. In addition, based on a strong similarity with ERa DNA binding domain (68%) but a moderate similarity (36%) with the ligand pocket of ER, precluding estrogen binding, three orphan nuclear receptors referred to as Estrogen receptor-Related Receptors (ERRs) ERRa, ERRb and ERRg have been described [START_REF] Audet-Walsh | The multiple universes of estrogen-related receptor α and γ in metabolic control and related diseases[END_REF]. Despite the identification of putative ligands, many data support the fact that ERRa impinges on the estrogen signaling pathway in numerous tissues including the bone (6) [START_REF] Bonnelye | An energetic orphan in an endocrine tissue: a revised perspective of the function of estrogen receptor-related receptor alpha in bone and cartilage[END_REF] [START_REF] Fujimoto | Estrogen-related receptor in reproductive organs[END_REF].

Skeletal metastases are frequent complications of many cancers of which prostate (PCa) and breast (BCa) cancers are the most frequent with a 73% and 68% incidence of bone metastases (BMet) [START_REF] Clézardin | Bone metastasis: mechanisms, therapies, and biomarkers[END_REF]. BMet development requires first cancer cell extravasation and homing to the bone marrow through interactions with endothelial cells and with osteoblasts (bone-forming cells).

Once in the bone, cancer cells can then disrupt the osteoblast/osteoclast balance; in favor of osteoclasts (bone-resorbing cells) or osteoblasts inducing osteolysis or osteoblastic/mixed lesions. Aside from bone cells, immune cells in the bone marrow (BM) also strongly influence BMet [START_REF] Mendoza-Reinoso | Contribution of Macrophages and T Cells in Skeletal Metastasis[END_REF]. In pre-clinical models, estrogen deficiency (OVX) has been shown to fuel BMet, since the declining production of the sex steroid by the ovaries and inflammatory tone associated with estrogen deficiency modified the bone microenvironment, mainly the osteoclasts, in such a way that cancer cell anchorage, survival and osteolytic phenotype were stimulated [START_REF] Wright | The microenvironment matters: estrogen deficiency fuels cancer bone metastases[END_REF]. Conversely in clinical studies, incidence of BM disseminated tumor cells (DTC) was reported to be slightly higher in pre-menopausal women (32,7%) versus postmenopausal (29,5%), suggesting that the BM of post-menopausal women is less attractive to metastasis [START_REF] Braun | A Pooled Analysis of Bone Marrow Micrometastasis in Breast Cancer[END_REF]. Of note, all immune cells present in the bone express the ERs [START_REF] Kovats | Estrogen receptors regulate innate immune cells and signaling pathways[END_REF], and growing evidence suggests a role for estrogens in BMet through their action on bone immune cells. Moreover, regarding basic science, it is important to keep in mind that a large part of the cultures of hematopoietic cells are carried out with media containing phenol red, known to have estrogenic activity. This review aims at presenting our current knowledge and our own thoughts on the links between estrogen signaling in bone immune cells and their impact on metastatic cell homing and progression within the bone/BM microenvironment.

1-Estrogen and myeloid cells

Aside from erythrocytes, myeloid cells, including mainly neutrophils, monocytes/macrophages and osteoclasts are the most abundant hematopoietic cells in the bone [START_REF] Zhao | Bone marrow and the control of immunity[END_REF]. All of these myeloid cells, the differentiation of which occurs in partially or completely in the BM, express not only ERa/b but also GPER1 [START_REF] Notas | G Protein-Coupled Estrogen Receptor in Immune Cells and Its Role in Immune-Related Diseases[END_REF]. We will see that estrogens affect both their differentiation and function in the bone with consequences on BMet niche, cancer cell homing and progression in the bone.

1-1-Neutrophils

Neutrophils largely outnumber the rest of the myeloid cells in the bone [START_REF] Von Vietinghoff | Homeostatic regulation of blood neutrophil counts[END_REF], where their numbers are influenced by estrogens. In the 80's, experiments based on the injection of estrogens to male mice revealed a profound neutropenia in the BM [START_REF] Crandall | Estrogens and hematopoiesis: characterization and studies on the mechanism of neutropenia[END_REF] [START_REF] Gaunt | Myelopoiesis and marrow adherent cells in estradiol-treated mice[END_REF]. Since then, it has been reported that estrogens impact both neutrophil differentiation and functions. Neutrophil differentiation is largely promoted by the Granulocyte-Colony-Stimulating Factor (GM-CSF).

In the absence of estrogen, B cells present in the BM secrete more G-CSF, which can contribute to the neutrophilia observed in estrogen-deficient patients [START_REF] Nordqvist | Effects of a tissue-selective estrogen complex on B lymphopoiesis and B cell function[END_REF]. Neutropenia in the bone associated with estrogens can also be explained by the ability of estrogens to down-regulate the production of CXC chemokine ligand (CXCL)12 by osteoblasts and BM stromal cells [START_REF] Ucer | The Effects of Aging and Sex Steroid Deficiency on the Murine Skeleton Are Independent and Mechanistically Distinct[END_REF]. Indeed, if a large portion of mature neutrophils leave the BM, the fraction expressing CXCR4, receptor of CXCL12, stays within the bone [START_REF] Ucer | The Effects of Aging and Sex Steroid Deficiency on the Murine Skeleton Are Independent and Mechanistically Distinct[END_REF]. These BM-resident neutrophils secrete Proteinase 3 (PR3), a serine protease which through its interaction with Receptor for Advanced Glycation Endproducts (RAGE) at the surface of metastatic PCa could enhance their homing to the bone [START_REF] Kolonin | Interaction between Tumor Cell Surface Receptor RAGE and Proteinase 3 Mediates Prostate Cancer Metastasis to Bone[END_REF]. Though only the depletion of neutrophils in vivo will firmly validate this conclusion, this observation strongly suggests that a large number of neutrophils in the bone could contribute to BMet incidence from PCa cells by encouraging PCa cell anchorage in the BM. One feature of neutrophils is that they form NETs (Neutrophil Extracellular Traps), which not only prevents pathogen progression but also, as recently demonstrated, contributes to the sequestring of liver and lung metastatic cells in the bone promoting their implantation [START_REF] Yang | DNA of neutrophil extracellular traps promotes cancer metastasis via CCDC25[END_REF].

Inhibition of ERa and ERb signaling in neutrophils by using selective antagonist raloxifene was associated with inhibition of NETs suggesting that estrogen promote NETs formation into the bone [START_REF] Flores | The Selective Estrogen Receptor Modulator Raloxifene Inhibits Neutrophil Extracellular Trap Formation[END_REF]. It is noteworthy that NET fails to attract primary cancer cells from breast to bone. This difference depending on metastases origin was associated with the ability of metastasis to express or not CCD25, which recognizes DNA chromatin filaments composing the NET [START_REF] Yang | DNA of neutrophil extracellular traps promotes cancer metastasis via CCDC25[END_REF].

Once settled in the bone, BMets originating from PCa DTC stimulate oxidative bursts in neutrophils, increasing NET formation. In turn, neutrophils induce apoptosis of PCa cells by inhibiting STAT5 signaling in cancer cells in a manner that remains to be understood [START_REF] Costanzo-Garvey | Neutrophils are mediators of metastatic prostate cancer progression in bone[END_REF].

Interestingly, this cytotoxic property of neutrophils decreases with time and in late-stage bone tumors. High levels of E2 also lead to a decrease in degranulation of β-glucuronidase and lysozyme, as well as in the intracellular concentration of reactive oxygen species (ROS) in neutrophils known to stimulate osteoclasts and osteoblast apoptosis (25)(26) [START_REF] Agidigbi | Reactive Oxygen Species in Osteoclast Differentiation and Possible Pharmaceutical Targets of ROS-Mediated Osteoclast Diseases[END_REF]. Hence, by controlling both neutrophil homeostasis and functions in the bone, estrogens repress the trapping of disseminated cells in the bone, but diminish the anti-tumor activity at an early stage and decrease the osteogenic feature associated with neutrophils once the tumor is settled within the bone (Figure1).

1-2-Monocytes-macrophages

In addition to neutropenia, estrogens also repress the differentiation of mononuclear phagocytic cells. Indeed, estrogens reduce the levels of Macrophage-Colony-Stimulating Factors (M-CSF), mainly produced by osteocytes, osteoblasts, and osteoblast precursors as well as Granulocyte-Macrophage Colony-Stimulating Factor (GM-CSF), this latter being secreted by multiple cell types in the bone mainly in response to danger stimuli (28)(29)(30) [START_REF] Riggs | The mechanisms of estrogen regulation of bone resorption[END_REF]. Bone macrophages, mostly arise through the differentiation of monocytes returning to the bone. In addition to repressing monocyte/macrophage lineage differentiation, estrogens also reduce concentrations of CCL2 (C-C Motif Chemokine Ligand 2) produced by osteoblasts and osteocytes, and thus the recruitment to the bone of monocytes and macrophage-precursors which expressed CCR2 [START_REF] Yang | Elevated CCL2/MCP-1 Levels are Related to Disease Severity in Postmenopausal Osteoporotic Patients[END_REF]. Estrogens also directly inhibit the production of CCL2 by monocyte/macrophage and thus the positive loop of macrophage recruitment to the bone [START_REF] Riggs | The mechanisms of estrogen regulation of bone resorption[END_REF]. Interestingly, depending on the ER engaged, estrogens can have different direct effects on monocyte/macrophage functions. In monocytes/macrophages, E2/ERa inhibits the production of pro-inflammatory cytokines, such as IL-1b, TNF-a, IL-6, documented to reduce osteoclast differentiation (31)(32) [START_REF] Buenrostro | Dissecting the role of bone marrow stromal cells on bone metastases[END_REF].

Conversely, E2/ERb represses CD16 surface expression and thus ADCC (antigen-dependent cellular cytotoxicity) [START_REF] Trenti | Estrogen, Angiogenesis, Immunity and Cell Metabolism: Solving the Puzzle[END_REF]. Whether different stages of macrophage maturation are more strongly associated with the expression of one type of ER than another remains to be addressed.

ERRa, whose expression is up-regulated by estrogens in several tissues including the bone (8) [START_REF] Liu | Estrogen stimulates estrogen-related receptor alpha gene expression through conserved hormone response elements[END_REF] [START_REF] Bonnelye | Estrogen receptor-related receptor alpha impinges on the estrogen axis in bone: potential function in osteoporosis[END_REF], is involved in macrophage functions. More precisely, it can regulate macrophage response to TLR4 and IFNg as well as ROS production (37)(38) [START_REF] O'neill | Immunometabolism governs dendritic cell and macrophage function[END_REF]. Through their potent phagocytic activity, macrophages represent powerful cells to eliminate metastatic cells reaching the bone, albeit they can also influence BMet through other mechanisms that can be targeted by estrogens. "Indeed, estrogens strongly influence macrophage polarization by promoting M2 (pro-osteoblastic) and suppressing M1 (pro-osteoclastic) (34) [START_REF] Dou | Estrogen Deficiency-Mediated M2 Macrophage Osteoclastogenesis Contributes to M1/M2 Ratio Alteration in Ovariectomized Osteoporotic Mice[END_REF]. Therefore, by promoting M1 macrophages estrogen deficiency inhibits BMP2 production by M2 and stimulate proosteoclastic molecules production ROS, nitric oxide, and pro-inflammatory cytokines by M1, promoting bone resorption [START_REF] Chen | Communications Between Bone Marrow Macrophages and Bone Cells in Bone Remodeling[END_REF] . If this balance in favor of M2 contributes to preventing the establishment of the metastatic niche in bone, once metastasis is implanted, M2-like macrophages promote metastasis angiogenesis and tumor progression through their production of VEGF-A [START_REF] Mendoza-Reinoso | Contribution of Macrophages and T Cells in Skeletal Metastasis[END_REF]. In the bone, a peculiar subset of macrophages named osteal macrophages, or osteomacs, has been described [START_REF] Michalski | Macrophages and skeletal health[END_REF]. Osteal macrophages (TRAP -F4/80 + , CD68 + , Mac3 + ) are present at the bone surface in the close vicinity of mature osteoblasts where they support osteoblastic function and bone anabolism [START_REF] Cho | Osteal macrophages support physiologic skeletal remodeling and anabolic actions of parathyroid hormone in bone[END_REF]. Data have connected estrogen levels and osteal macrophage functions, since the number of osteal macrophages increased on both trabecular and endocortical bone post-ovariectomy. Interestingly, in this condition, osteal macrophages contain TRAP + intracellular vesicles, attesting to their "clean up" role of exocytosed postresorption vesicles released by osteoclast (TRAP + ) activity [START_REF] Batoon | Osteal macrophages support osteoclast-mediated resorption and contribute to bone pathology in a postmenopausal osteoporosis mouse model[END_REF]. PCa-derived BMet increases osteoblast functions through osteoblastic production of CCL2, which favors osteal macrophage positioning [START_REF] Siddiqui | GDF15 promotes prostate cancer bone metastasis and colonization through osteoblastic CCL2 and RANKL activation[END_REF]. Tumor-associated macrophages (TAM) have been proposed potentiate BCa and PCa BMet. Increased numbers of CD206 + M2-like macrophages have also been found in PCa BMet [START_REF] Lo | Multifaceted Roles for Macrophages in Prostate Cancer Skeletal Metastasis[END_REF] [START_REF] Soki | Bone marrow macrophages support prostate cancer growth in bone[END_REF]. CCL2 expression by cancer cells promotes the recruitment of TAM expressing CCR2 to facilitate cancer cell anchorage in the bone [START_REF] Buenrostro | Dissecting the role of bone marrow stromal cells on bone metastases[END_REF]. In the same line, the expression of Parathyroid Hormone-Related Protein (PTHRP) by BCa and PCa cells upregulates the production of CCL2 in osteoblasts that contribute to macrophage recruitment to the bone, bone remodeling and BMet progression [START_REF] Mulholland | Monocyte Chemoattractant Protein-1 (MCP-1/CCL2) Drives Activation of Bone Remodelling and Skeletal Metastasis[END_REF] [START_REF] Li | A destructive cascade mediated by CCL2 facilitates prostate cancer growth in bone[END_REF]. Moreover, the clearance of tumor apoptotic cells by efferocytosis promotes CXCL5 production by macrophages and an inflammatory bone microenvironment supporting BMet development [START_REF] Roca | Apoptosis-induced CXCL5 accelerates inflammation and growth of prostate tumor metastases in bone[END_REF]. Thus, estrogens repress the number of monocytes/macrophages in the bone including osteal macrophages and affect their functions. Estrogens enhance their non-inflammatory cytokine production and repress bone remodeling, preventing formation of a fertile soil for BMet anchorage. However, once BMet manage to settle in the bone, estrogens provide through their action on macrophages a microenvironment facilitating tumor progression (Figure1).

1-3-Osteoclasts

Osteoclasts are also derived from the monocyte/macrophage lineage. However, on inflammatory conditions, osteoclasts can arise from either dendritic cells or erythromyeloid progenitors (51)(52)(53) [START_REF] Yahara | Erythromyeloid progenitors give rise to a population of osteoclasts that contribute to bone homeostasis and repair[END_REF]. In addition to the indirect effects described above of estrogens on osteoclast lineage differentiation, selective ablation of ERa in osteoclasts was reported to induce trabecular bone loss in female mice due to decreased FasL expression and autocrine regulation, leading to the inhibition of osteoclast apoptosis [START_REF] Nakamura | Estrogen prevents bone loss via estrogen receptor alpha and induction of Fas ligand in osteoclasts[END_REF]. As for macrophages/monocytes, osteoclasts are also antigen-presenting cells able to activate CD4 + T cells and CD8 + T cells, and are endowed with a unique ability to induce Foxp3 + regulatory T cells (Treg) (56) [START_REF] Kiesel | Cross-presentation by osteoclasts induces FoxP3 in CD8+ T cells[END_REF]. Moreover, osteoclasts produce a number of chemokines (Cc13, Cc14, Ccl5, Cxcl5, Cxl10, Cxcl11) attracting multiple immune cells e.g. monocytes-macrophages, Tcells, NK and DC cells [START_REF] Kiesel | Systems level analysis of osteoclastogenesis reveals intrinsic and extrinsic regulatory interactions[END_REF]. PCa-derived BMet increase osteoblast functions and facilitate their growth in bone by activating osteoclastogenesis through osteoblastic production of the Receptor Activator of Nuclear factor Kappa-B ligand (RANKL), a key factor for osteoclast differentiation [START_REF] Siddiqui | GDF15 promotes prostate cancer bone metastasis and colonization through osteoblastic CCL2 and RANKL activation[END_REF]. ERRa is also a strong regulator of osteoclast differentiation (59)(60) [START_REF] Wei W | Ligand Activation of ERRα by Cholesterol Mediates Statin and Bisphosphonate Effects[END_REF].

Interestingly, depending on the metastasis origin, overexpression of ERRa in cancer cells either inhibits or stimulates osteoclasts by stimulating the expression of either the decoy receptor of RANKL, OPG (osteoprotegerin), a major inhibitor of osteoclast activity in BCa cells, or VEGF-A and WNT5a in PCa cells (62) [START_REF] Fradet | Estrogen related receptor alpha in castration-resistant prostate cancer cells promotes tumor progression in bone[END_REF]. Now the impact of estrogens on ERRa in oncology and their role on BMet modulation remains to be determined. Through their action on osteoclasts, estrogens repress bone resorption but also modify the ability of osteoclasts to interact and recruit other immune cells to the bone with direct consequences on the development of a proinflammatory microenvironment in favor of BMet (Figure1).

2-Estrogen and lymphoid cells

2-1-T cells

T lymphocytes present in the BM account for less than 5% of CD45 + cells and have a reduced CD4/CD8 ratio compared to the blood. The BM T cell compartment is almost exclusively composed of a,b T lymphocytes [START_REF] Hillhouse | Immunoregulatory CD4(-)CD8(-) T cells as a potential therapeutic tool for transplantation, autoimmunity, and cancer[END_REF]. A large part of the T cells present in the BM are memory cells, which either circulate or permanently inhabit the BM, suggesting that in the context of BMet, BM T cells could compose a pool of T cells with specificity against antigens bared by cancer cells [START_REF] Beckhove | Specifically activated memory T cell subsets from cancer patients recognize and reject xenotransplanted autologous tumors[END_REF]. The entry of T cells into the BM is largely supported by CXCL12 produced by osteoblasts and stromal cells, and down-regulated by estrogens [START_REF] Ucer | The Effects of Aging and Sex Steroid Deficiency on the Murine Skeleton Are Independent and Mechanistically Distinct[END_REF]. In the absence of intravital/in situ labeling data, there is so far no direct evidence that estrogens affect T cell retention within the bone. However, this phenomenon is largely dictated by the interaction between a4b1 integrin, present at the surface of T cells, and VCAM-1 expressed by stromal cells and endothelial cells and depicted to be down-regulated by estrogens (66) [START_REF] Ulyanova | VCAM-1 expression in adult hematopoietic and nonhematopoietic cells is controlled by tissueinductive signals and reflects their developmental origin[END_REF]. One of the main roles of estrogens on BM T cells is to repress their osteoclastogenic ability. Evidence of this function emerged more than 20 years ago, after the observation that in contrast to wild type mice, nude mice are protected from trabecular bone loss induced by ovariectomy (68) [START_REF] Cenci | Estrogen deficiency induces bone loss by enhancing T-cell production of TNF-alpha[END_REF].

Later, CD4 T cells producing IL-17 and RANK-L in the bone were identified as potent stimulators of osteoclastogenesis [START_REF] Okamoto | Osteoimmunology: The Conceptual Framework Unifying the Immune and Skeletal Systems[END_REF]. More recently a Th17 osteoclastogenic T cell subset expressing high amounts of TNF-a, and established in the gut, was proposed to reach the bone in both mice and humans (71) [START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF], in a CCL2-dependent manner [START_REF] Uehara | Current perspectives of osteoclastogenesis through estrogen modulated immune cell cytokines[END_REF]. Thanks to their osteoclastogenic ability Th17 cells have been suggested to establish a bone pre-metastatic niche, which facilitates cancer cell implantation in the bone (74) [START_REF] Monteiro | CD8+ T cells from experimental in situ breast carcinoma interfere with bone homeostasis[END_REF]. In addition, Th17 cells were suggested to preclude BMet control anti-PD1 treatment [START_REF] Jiao | Differences in Tumor Microenvironment Dictate T Helper Lineage Polarization and Response to Immune Checkpoint Therapy[END_REF]. Estrogens directly control the Th17 cell pool since ERa binds the promoter of RORc inhibiting Rorgt expression (77) [START_REF] Tyagi | Estrogen Deficiency Induces the Differentiation of IL-17 Secreting Th17 Cells: A New Candidate in the Pathogenesis of Osteoporosis[END_REF].

In addition to Th17 cell differentiation, estrogens repress the Th1 cell program (79). In the BM, estrogen deficiency impairs the production of TGF-b by stromal cells, a cytokine that plays a key role in the repression of T-Bet and IFN-g/TNF-a expression in T cells [START_REF] Marie | Cellular mechanisms of fatal early-onset autoimmunity in mice with the T cell-specific targeting of transforming growth factorbeta receptor[END_REF] and thus in osteoclastogenesis [START_REF] Gao | Estrogen prevents bone loss through transforming growth factor beta signaling in T cells[END_REF]. Given the potent role of TGF-b in repressing the expression of granzyme-A,-B, and FasL in CD8 + T cells [START_REF] Thomas | TGF-beta directly targets cytotoxic T cell functions during tumor evasion of immune surveillance[END_REF], the ability of estrogens to sustain TGF-b levels in the BM impairs CD8 + T cell cytotoxic functions, and promotes BMet progression [START_REF] Janssen | The immune system in cancer metastasis: friend or foe?[END_REF].

However, the control by estrogens/estrogen signaling of TGF-b levels in bone and in BMet is more complex. Indeed, bone contains a large source of TGF-b stored in its mineralized matrix, which is released and activated by osteoclast activity. In addition, BMet can also contribute to increasing the levels of TGF-b [START_REF] Trivedi | The Role of TGF-β in Bone Metastases[END_REF]. ERRa is also clearly involved in effector T cell activation (85) [START_REF] Liebmann | Nur77 serves as a molecular brake of the metabolic switch during T cell activation to restrict autoimmunity[END_REF]. In BCa BMet, ERRa expression restrains TGF-b production by cancer cells, leading to exacerbated cytotoxic features in CD8 + T cells in the bone and efficient BMet control [START_REF] Bouchet | ERRα expression in bone metastases leads to an exacerbated anti-tumor immune response[END_REF].

TGF-b has been associated with an increase in the generation and stability of Tregs [START_REF] Marie | TGF-beta1 maintains suppressor function and Foxp3 expression in CD4+CD25+ regulatory T cells[END_REF]. In the BM, the proportion of Tregs among CD4 + T cells is much higher than in the lymph nodes, likely due to high levels of CXCR4 expression on Tregs (89) largely promoted by estrogens [START_REF] Fan | Lack of estrogen downregulates CXCR4 expression on Treg cells and reduces Treg cell population in bone marrow in OVX mice[END_REF].

Once the cancer cell is anchored, Tregs produce high amounts of RANKL promoting osteolysis associated with a feedback loop of TGF-b release in BCa BMet [START_REF] Tan | Tumourinfiltrating regulatory T cells stimulate mammary cancer metastasis through RANKL-RANK signalling[END_REF].

Hence, in the bone, the effects of estrogens on T cells are dual: by contributing to an immunesuppressive environment associated with reduced activity of CD8 + T cells and increased number of Tregs which can favor tumor growth and by limiting osteolysis due to inhibition of RANKL production by Th1 and Th17 cells (Figure1).

2-2 B cells

BM is the main site of B lymphopoiesis and maturation. In mice, estrogen treatment was linked to a decrease in the numbers of pro/pre-(B220 low /IgM -) and immature (B220 low /IgM +) B subpopulations, whereas the mature (B220 high /IgM + ) subpopulation increased in the BM [START_REF] Thurmond | Role of estrogen receptor alpha in hematopoietic stem cell development and B lymphocyte maturation in the male mouse[END_REF]. This modification of B cell precursor homeostasis does not seems to be associated with a loss of survival of pro/pre-and immature B cells since estrogens induce an increase in the antiapoptotic factor Bcl2 in these cells [START_REF] Grimaldi | Estrogen alters thresholds for B cell apoptosis and activation[END_REF]. The exact mechanism of action of estrogens on B lymphopoiesis in vivo remains unknown. However, co-culture approaches suggested that estrogens modify the ability of BM stromal cells to support efficient B lymphopoiesis [START_REF] Masuzawa | Estrogen deficiency stimulates B lymphopoiesis in mouse bone marrow[END_REF].

Estrogens promote the production of soluble Frizzled-related protein 1 by BM stromal cells, which in turn stabilizes β-catenin and blocks early B lymphoid progression [START_REF] Yokota | Soluble frizzled-related protein 1 is estrogen inducible in bone marrow stromal cells and suppresses the earliest events in lymphopoiesis[END_REF]. Of note, B cell interactions with other immune cells in the BM conditions bone homeostasis. Through CD40/CD40L interaction, B cells and T cells cooperate to sustain normal bone mass and mineral density. Signals delivered by CD40 engagement stimulates in B cells the production of OPG and consequently reduce osteoclast differentiation. Interestingly, in post-menopausal women, a switch from OPG to RANK-L production is observed in B cells emphasizing bone loss [START_REF] Fujiwara | RANKL (Receptor Activator of NFκB Ligand) Produced by Osteocytes Is Required for the Increase in B Cells and Bone Loss Caused by Estrogen Deficiency in Mice[END_REF] [START_REF] Eghbali-Fatourechi | Role of RANK ligand in mediating increased bone resorption in early postmenopausal women[END_REF]. The role of B cells in the direct control of BMet seems elusive, since in mice with BMet, as in humans, mature B cells are scattered over the BM close to osteoclasts rather than metastasis [START_REF] Li | Distribution of macrophages, osteoclasts and the B-lymphocyte lineage in osteolytic metastasis of mouse mammary carcinoma[END_REF]. However, by controlling B lymphopoiesis, and modifying mature B cell profile in the BM, estrogen levels modify bone homeostasis by reducing osteolysis and thus may limit cancer cell anchorage and progression in the bone.

2-3 Innate lymphoid cells (ILCs)

In the adult, the BM is regarded as the main site for ILC-poiesis. Several multipotent ILC precursors have been defined in mice with different degrees of pluripotency that are still under debate [START_REF] Diefenbach | Development, differentiation, and diversity of innate lymphoid cells[END_REF]. Among ILCs, ILC-1/Natural killer (NK) cells express both ERa/b and are prevalent in the BM. In contrast, outside the uterus ILC-2 fails to express ERs, and ILC-3 are endowed with a clear tropism for the mucosa (100). If the effects of estrogen on NK cell ontogeny in the BM are not well documented, there is more evidence of a role, either direct or indirect, for estrogens on NK functions in BM. Interestingly, both development and activation of BM NK cells are dependent on IL-15, a cytokine also important for osteoclast development (101) and abundantly produced by a fraction of mesenchymal cells named CAR cells (CXCL12-abundant reticular) (102)(103), the functions of which are orchestrated by estrogens.

Once NK cells have matured, the down-regulation of CXCR4 and up-regulation of S1P5 drive their egress from the CXCL12 enriched BM microenvironment (104). A large population of resident CXCR6 + CD69 + NK cells is observed but seems to be endowed with a weak cytotoxic ability against cancer cells, whereas the major anti-tumor activity of NK is supported by recirculating mature NK cells (105). Estrogens reduce the cytotoxic activity (production of granzymes and FasL) of mature NK cells, likely through the down-regulation of their activation receptors (NKp46, NKG2D/L) and the increased expression of the inhibiting receptor CD94 (106). However in the bone, no effect of NK on BMet progression has clearly been documented (107). Hence, the reduced cytotoxic function of NK cells in response to estrogens likely increases the number of disseminated cells in the blood that could reach the bone, but do not contribute to the metastasis immune escape once settled in the bone. However activated NK cells can either promote or inhibit osteoclastogenesis depending on the release of TNF-a or IFN-γ, respectively and this regulation is influenced by estrogens which repress IFNg in NK cells (106)(108)(109). Activated NK cells can also directly lyse mature osteoclasts (110).

Conversely, zoledronate treatment, the most potent bisphosphonate for the treatment of BMet, protects osteoclasts from NK cell cytotoxicity (110). Thus, estrogens induce a large repression of NK cell functions in the bone, which in the light of the ability of NK cells to regulate bone resorbing cells could impact bone remodeling and BMet progression (Figure1).

Conclusion

The last decades of works on estrogens and BM brought to light the fact that estrogens affect the immune compartment of the bone and possibly the global system. Therefore, this observation may explain why adult females mount stronger immune responses than males, and strongly questions the consequences of modifications in estrogen levels on innate and adaptive immune responses during aging and gender reassignment. Moreover, the emergence of the immunomodulatory role of estrogens suggests that anti-estrogenic drugs, or estrogen administration, by injection or topical applications, may influence the BM. Subsequently this treatment may have an indirect anti-tumor effects by modulating the immune system and may therefore participate to the modulation of pre-metastatic niches and/or modulate the tumor immune environment in the bone. Anti-estrogenic/and estrogenic therapies may also modify the ability of patients to respond to immunotherapy by affecting BM immune cell development and function. This area needs to be particularly explored given the increasing number of people either treated with anti-estrogenic drugs or receiving estrogen.

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Acknowledgment

This work was supported by the National Center for Scientific Research (CNRS) to EB, the National Institute of Health and Medical Research (INSERM) to JCM, by the Labex DevWECan ANR-10-LABX-6 (JCM). The SIRIC LYriCAN INCa-DGOS-inserm_12563 is acknowledge. We thank Brigitte Manship for the editing.


Andrew Ferguson

Giorgio Ottaviani

Mohab Safey

El Din

Ettore Turatti

On the degree of varieties of sum of squares

Keywords:

We study the problem of how many different sum of squares decompositions a general polynomial f with SOS-rank k admits. We show that there is a link between the variety SOS k (f ) of all SOSdecompositions of f and the orthogonal group O(k). We exploit this connection to obtain the dimension of SOS k (f ) and show that its degree is bounded from below by the degree of O(k). In particular, for k = 2 we show that SOS 2 (f ) is isomorphic to O(2) and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank k and obtain the degree in the special case k = 2.

Introduction

Motivation. Let V be a complex vector space of dimension n + 1 with basis {x 0 . . . , x n } and let d ≥ 0 be an integer. Let f ∈ C[x 0 , . . . , x n ] be a homogeneous polynomial of degree 2d, that is f ∈ Sym 2d V . A starting case, when f is real, is the problem of computing the global infimum of f , f * = inf z∈R n f (z). Polynomial optimisation problems appear frequently in practice in many different fields, including areas of engineering and social science such as computer vision [START_REF] Probst | Convex Relaxations for Consensus and Non-Minimal Problems in 3D Vision[END_REF][START_REF] Aholt | A Hilbert Scheme in Computer Vision[END_REF], control theory [START_REF]Positive Polynomials in Control[END_REF][START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF] and optimal design [DCGH + 19]. However, even for deg f ≥ 4 this is an NP-hard problem [START_REF] Katta | Some NP-complete problems in quadratic and nonlinear programming[END_REF]. As such, many methods have been developed to approximate f * . A popular method is to relax the optimisation problem:

max λ∈R λ s.t. f -λ = k i=1 g 2 i , g i ∈ Sym d V.
Clearly, being a sum of squares implies non-negativity. It is well-known that these notions are equivalent in two homogeneous variables. However, due to the counter example by Motzkin this is not true in general [START_REF] Motzkin | The arithmetic-geometric inequality[END_REF].

In [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], using the duality between moments and sums of squares, Lasserre constructed a hierarchy of semi-definite programs whose solutions converge to the true infimum f * . However, in general, the decompositions obtained from semi-definite programming are approximate certificates of non-negativity. In recent years there has been an increased study on computing exact certificates [START_REF] Peyrl | Computing sum of squares decompositions with rational coefficients[END_REF][START_REF] Magron | On exact Polya and Putinar's representations[END_REF]. Hence, one wants to understand the algebraic structure of SOS decompositions and the related semi-definite programs.

Prior works. Following the classical works of Sylvester [START_REF] Joseph | On a remarkable discovery in the theory of canonical forms and of hyperdeterminants[END_REF], the study of so-called Waring decompositions, decompositions of homogeneous polynomials by powers of linear forms, is an active area of research. In [START_REF] Fröberg | On the Waring problem for polynomial rings[END_REF] it was proved that any general f ∈ Sym 2d V is a sum of at most 2 n squares. For fixed n, this bound is sharp for all sufficiently large d. The authors of [START_REF] Lundqvist | On generic and maximal k-ranks of binary forms[END_REF] investigate the minimal numbers of squares in a decomposition of a generic polynomial in two variables. Then, in [START_REF] Fröberg | Algebraic stories from one and from the other pockets[END_REF], the authors give a conjecture on the generic SOS-rank of polynomials, see Definition 1.2, in terms of number of variables and degree. On the other hand, in this paper we will study generic polynomials of a given SOS-rank.

In this paper, one aim is to analyse the degree of SOS decompositions directly from an algebraic geometry point of view. Another aim is to better understand the structure of the SOS decompositions of a given polynomial.

Main results. We consider SOS decompositions of polynomials of degree 2d.

Definition 1.1. Let f ∈ Sym 2d V . The polynomial f has SOS-rank k if k is the minimum number such that there exist

f i ∈ Sym d V such that f = r i=1 f 2 i .
In this paper, we define and study two varieties related to exact SOS decompositions. The first is defined by all polynomials of rank less than or equal to k, with a general point f ∈ SOS k being a polynomial of rank k.

Definition 1.2. Let SOS k be the subvariety in Sym 2d V obtained from the Zariski closure of the set of all SOS-rank k polynomials.

SOS k = {f 2 1 + • • • + f 2 k | f i ∈ Sym d V }.
The generic SOS-rank is the smallest number k such that SOS k covers the ambient space.

Another notion that can be explored is instead of analysing all polynomials of a given rank, one can seek to understand all the different decompositions of the general polynomial f . Definition 1.3. Let f ∈ SOS k be a generic polynomial. We define the variety of all the SOS decompositions of f as

SOS k (f ) = {(f 1 , . . . , f k ) ∈ k i=1 Sym d V | k i=1 f 2 i = f }.
While we investigate the SOS k (f ) variety for all ranks k, in particular we give a complete description of the k = 2 case.

Theorem 1.4. Let f ∈ SOS 2 be a generic polynomial that is the sum of two squares. Then, SOS 2 (f ) has two irreducible components isomorphic to SO(2).

Since SO(k) acts on any decomposition using k squares, we have the inequality

dim SOS k (f ) ≥ dim SO(k) = k 2 .
In Corollary 3.6 we prove a statement which implies the following result.

Theorem 1.5. Let f ∈ SOS k be generic with k ≤ n. Then,

dim SOS k (f ) = k 2 .
By analysing the general polynomial in SOS 2 , we prove a formula for the degree of this variety.

Theorem 1.6. Let N = dim Sym d V = n+d d . The degrees of the varieties of squares and of sum of two squares in P(Sym 2d V ) are given by

deg(SOS 1 ) = 2 N -1 , deg(SOS 2 ) = N -3 i=0 N +i N -2-i 2i+1 i
.

Moreover, the dominant map π :

k i=1 Sym d V → SOS k (f 1 , . . . , f k ) → k i=1 f 2 i has fibers π -1 (f ) = SOS k (f )
, so that Theorem 1.5 implies the following.

Corollary 1.7.

dim SOS k ≤ k n + d n - k 2
and equality holds for k ≤ n and a general f ∈ SOS k .

Structure of the paper. In Section 2, we begin by recalling some definitions in sums of squares decompositions, algebraic geometry and commutative algebra. Then, in Section 3 we investigate the variety of all possible sums of k-squares decompositions of a given polynomial. We describe the action of the orthogonal group of size k on this variety and conjecture that there is an isomorphism between these two objects. We provide experimental and theoretical support for this conjecture and conclude by showing that it holds for k = 2. Finally, in Section 4 we use the results of Section 3 to prove a formula for the degree of the variety of all SOS decompositions of two squares in addition to a upper bound on this degree for k ≥ 3.

Preliminaries

Let V be a complex vector space of dimension n + 1. We will denote the n-dimensional projective space associated to V by PV .

Definition 2.1. We define the d-Veronese embedding as the map

ν d : PV → P Sym d (V ), → d .
Notice that the map ν d is closed [START_REF] Shafarevich | Basic Algebraic Geometry 1: Varieties in Projective Space[END_REF]. Therefore, we define the d-Veronese variety in P Sym d (V ) as the the image of PV under the Veronese embedding

ν d . Definition 2.2. A polynomial f ∈ Sym d V has rank one, or is decomposable, if f = v d . The rank of a polynomial f is defined as the minimum number r ∈ N such that f = r i=1 v d i .
In other words, f is the sum of r decomposable polynomials.

Observe that the Veronese variety ν d (V ) ⊂ P Sym d V consists exactly of the rank one polynomials.

Definition 2.3. Let X be a subvariety of V . The k-th secant variety of X, denoted Σ k (X), is defined as the Zariski closure of the union of all the k linear subspaces spanned by points in X. That is

Σ k (X) = x1,...,x k ∈X span{x 1 , . . . , x k }.
If X = ν d (PV ) ⊂ P Sym d V , then the generic elements in the k-th secant variety of the Veronese variety consist exactly of polynomials of rank k as long the inclusion Σ k (ν d (PV )) ⊂ P Sym d V is strict.

Let U denote Sym d V . We can decompose Sym 2d U follows:

Sym 2 U = Sym 2d V ⊕ C,
where C is obtained by plethysm, see [START_REF] Weyman | Cohomology of Vector Bundles and Syzygies[END_REF] for more details. The space C corresponds to the quadrics on U that vanish on ν d (PV ). Moreover, Sym 2d V is the degree two piece of the coordinate ring of ν d (PV ).

Let {x 0 , . . . , x n } be a basis of V . Consider a basis

w 1 = x d 0 , w 2 = x d-1 0 x 1 , . . . , w N = x d n , with N = n+d d . A rank one quadric q in Sym 2 U has an expression q = (α 1 w 1 + • • • + α N w N ) 2 , with α 1 , . . . , α N ∈ C.
Switching to the coordinates given by V we have

q = (α 1 x d 0 + • • • + α N x d n ) 2 .
This means that rank one quadrics in Sym 2 U correspond to square powers in Sym 2d V . Furthermore, applying the same argument for a rank k quadric f ∈ Sym 2 U , we see that f corresponds to a sum of k squares in Sym 2d V . Notice that if (f 1 , . . . , f k ) ∈ SOS k (f ), as defined in Definition 1.3, then for any permutation σ ∈ S k , where S k is the symmetric group of order k, we have that

f σ(1) , . . . , f σ(k) ∈ SOS k (f ).
One could desire to remove such "overlapping" points by taking the quotient by S k . However, there is another important group, containing such permutations, that acts on SOS k (f ).

Let O(k) be the orthogonal group of order k. Fix a point (f 1 , . . . , f k ) ∈ SOS k (f ) and fix the ordering of the basis {w 1 , . . . , w N } of Sym d V . Define A to be the k × N matrix whose i-th row is the coefficients of the polynomial f i . Then,

xA t Ax t = f.
Let O ∈ O(k), then the action on the left by A preserves the polynomial f . That is,

x(OA) t OAx t = f.
Essentially, such an action leads to a different decomposition (f 1 , . . . , f k ) of f , where f i is the ith row of the matrix OA.

Definition 2.4. Let f ∈ Sym d V , let {x 0 , . . . , x n } be a basis of V and let ∂ 0 , . . . , ∂ n be the dual basis of V ∨ . For each m < d, we define the linear map

W m f :Sym m V ∨ → Sym d-m V, ∂ i1 • • • ∂ im → ∂f ∂x i1 • • • ∂x im .
The matrix corresponding to this linear map is called the catalecticant matrix of f .

We give some cohomological definitions that are going to be used later on. Let S = q Sym q (V ) be the symmetric algebra of V . Definition 2.5. Let R be a ring and F a free module of rank r over R. Given an R-linear map k : F → R, the complex

0 → r F ϕr --→ r-1 F ϕr-1 ---→ . . . ϕ2 -→ F ϕ1 -→ R → 0
is called the Koszul complex associated to k. The maps ϕ l are defined as

ϕ l (e 1 ∧ • • • ∧ e ) = i=1 (-1) i+1 k(e i )e 1 ∧ • • • ∧ e i ∧ • • • ∧ e ,
where the notation e i means that this element is omitted from the product. Definition 2.6. Let M be a finitely generated graded S-module and let F 0 , . . . , F m be the free S-modules that give a minimal free resolution of M . That is, there is an exact sequence

0 → F m → F m-1 → • • • → F 1 → F 0 → M → 0,
and the matrices of the maps φ i : F i+1 → F i have no non-zero constant entry, see [START_REF] Eisenbud | Commutative Algebra: With a View Toward Algebraic Geometry[END_REF]. The Betti number β i,j is the number of generators of degree j needed to describe F i . That is, F i = j S(-j) βi,j , where S(-j) is the j-graded part of S.

Definition 2.7. Let M , N be two graded S-modules and let F • be a free resolution of N . Consider the complex F • ⊗ M . The Tor groups are defined by

Tor S p (M, N ) = H p (F • ⊗ M ).
The next result shows the relation between the Tor groups of M and the Betti numbers of a free resolution of M .

Proposition 2.8. [Gre89, Section 1] Let m ⊂ S be the maximal ideal m = q≥1 Sym q (V ) and let k = S/m be the residual field. Then, Tor S p (M, k) q has rank equal to β p,q . This connection between the Betti numbers and the Tor groups is important because it correlates the Betti numbers with cohomology. This allows us to use semi-continuity on the Betti numbers, as explained in the next theorem.

Theorem 2.9. [Har77, Theorem 12.8] Let f : X → Y be a projective morphism of noetherian schemes. Let F be a coherent sheaf on X and flat over Y , in other words, F is a finitely presented O X -module and the functor -⊗ O Y,f (x) : Mod Fx → Mod Fx is exact for every x ∈ X. Then for each i ≥ 0, the function

y → dim H i (X y , F y )
is upper semi-continuous on Y .

The degree of the variety of all SOS decompositions

Let f = k i=1 f 2 i ∈ C[x 0 , . . . ,
x n ] be a sum of squares with degree 2d. We consider the variety in the ambient space

k i=1 Sym d (V ) of all possible SOS decompositions of the given polynomial f . SOS k (f ) = {(f 1 , . . . , f k ) ∈ k i=1 Sym d (V )| k i=1 f 2 i = f }
We conjecture the degree of this variety, when n ≥ k, to be the degree of the orthogonal group O(k).

In [BBB + 17] the authors give the degree of SO(k), and thus O(k), to be the determinant of the following binomial matrix

deg O(k) = 2 k det 2k -2i -2j k -2i 1≤i,j≤ k 2 . ( 1
)
For the case d = 1, the argument simplifies and so we give the following lemma.

Lemma 3.1. Let f ∈ Sym 2 V be a quadric of SOS-rank k ≤ n. Then, in the affine setting, the degree of SOS k (f ) is equal to the degree of O(k).

Proof. With f = k i=1 f 2 i , n ≥ k implies that we can encode f in a k × (n + 1) matrix, A, whose rows give the coefficients of the linear forms f i . Then, with x = (x 0 , . . . , x n ) we have that Ax t 2 = f . Thus, for any orthogonal matrix O ∈ O(k) we have that

OAx t 2 = (OAx t ) t (OAx t ) = xA t O t OAx t = xA t Ax t = Ax t 2 = f
Hence, there is an action on the SOS k (f ) variety by O(k). Additionally, there are at least two identical irreducible components that correspond to det O = ±1.

We now show that up to a change of coordinates and multiplication by an orthogonal matrix, this SOS expression is unique. Let A and B be k × (n + 1) matrices encoding SOS decomposition of f . Then, up to a change of coordinates, we can ensure that the first k columns are linearly independent and so QR decompositions can be found. Thus, let

A = Q 1 R 1 and B = Q 2 R 2 where Q 1 , Q 2 are k × k orthogonal matrices and R 1 , R 2 are k × (n + 1
) upper triangular matrices. Then, R 1 and R 2 also encode SOS decompositions of f . By the equation R 1 x t 2 = f we can identify exactly the entries of R 1 , up to multiplication by ±1 in the rows, or in other words, up to multiplication by an orthogonal matrix. The same holds for R 2 and so the decompositions encoded by A and B must be in the same orbit of the action of O(k) on SOS k (f ). Therefore, there is only one orbit and so the degree of SOS k (f ) is equal to the degree of O(k).

The argument above also works partially for the case d ≥ 2. Once a basis is chosen for Sym d V , we can construct the matrix in the same way with k rows but n+d d columns. Then, the group O(k) acts on the left to give new decompositions. However, the QR decomposition no longer implies uniqueness of the orbit. This is because there exist relations between the monomials described by the columns of f . In other words, the Gram matrix associated to f is not only symmetric but also has a moment structure. Thus, it is no longer easy to see that the non-linear equations given by the norm of Ax t squared, Ax t 2 , have a unique solution.

Experimentally, up to k ≤ 6, we observe a stabilisation of the degree of the variety SOS k (f ) as the degree of f increases. The following table derives from [BBB + 17, Table 1]. Since the degree of SOS 7 (f ) is at least 233, 232 for a generic f ∈ SOS 7 , k ≤ 6 is the currently the limit for our experimental methods.

k Symbolic Formula (O(k)) Formula (SO(k)) The next example shows that the condition n ≥ k is sharp.

2
Example 3.2. The general plane quartic can be expressed as g 2 1 + g 2 2 + g 2 3 in 63 ways, where g i ∈ Sym 2 C 3 . A proof of such result is presented in [Dol12, Theorem 6.2.3]. The idea is to consider the quartic form as the determinant of a 2 × 2 matrix whose entries are quadric forms.

The next lemma gives an indication of the connection between k-SOS decompositions and the orthogonal group O(k). Indeed, fixing a matrix A 0 ∈ M k×N is equivalent to fixing a sum of squares decomposition of rank k of f = x t A T 0 A 0 x.

Lemma 3.3. Let N ≥ k ≥ 1 be integers and A, A 0 ∈ M k×N be matrices, A 0 of maximal rank and consider the entries of A as variables x ij . Then the variety Y defined by the equation

A t A = A t 0 A 0 is isomorphic to O(k).
Proof. Up to an action of the group of N × N invertible matrices, GL(N ), on the left and O(k) on the right of A 0 , we may suppose without loss of generality that A 0 = I k 0 , with I k the k × k identity matrix and 0 a null matrix of size k × (N -k). Let A = X 0 X 1 , again with X 0 a k × k matrix and

X 1 a k × (N -k) matrix.
In those coordinates, the variety is determined by

X t 0 X 0 X t 0 X 1 X t 1 X 0 X t 1 X 1 = I k 0 0 0 . The first block implies that X t 0 X 0 ∈ O(k). Moreover X t 0 X 1 = 0 implies that X 1 = 0 since X 0 is invertible. Let f ∈ Sym 2d V be a sum of k squares f = k i=1 f 2 i . Then, f = xA t Ax t ,
where A ∈ M k×N has the coefficients of f i as its i-th row. This gives a natural isomorphism

SOS k (f ) ∼ = {B ∈ M k×N |xB t Bx t = f }.
(2)

Denote the Gram matrix W A = A t A and note that rk W A = k when the above decomposition is minimal. The previous lemma implies that O(k) ∼ = {B ∈ M k×N |W B = W A } ⊂ SOS k (f ). This, together with the isomorphism (2) implies that SOS k (f ) can be described by as many copies of O(k) as the number of distinct symmetric matrices W A of rank k such that x t W A x = f . Let f ∈ Sym 2d V and N = n+d d . Notice that the following diagram commutes.

C k ⊗ C N Sym 2 (C N ) Sym 2d V ϕ A -→ A t A A →xA t Ax t π B -→ xBx t (3) We have that SOS k (f ) = {A|π(A t A) = f }. Moreover, if B ∈ Im ϕ then rk B ≤ k.
The fiber π -1 (f ) = W 0 +C, where W 0 is the rank N catalecticant matrix of f such that xW x t = f and C is the variety

C = {C 0 ∈ S N | x T C 0 x = 0}.
This means that the problem can be reformulated in terms of the intersection ϕ(C k ⊗C N )∩(C+W 0 ): when this intersection is just a single point, as is the case for k ≤ 6 shown in Table 1, this implies that there exists only one C 0 ∈ C such that W 0 + C 0 has rank k. This is equivalent to saying that SOS k (f ) consists of a single copy of O(k). Thus, we arrive at the following conjecture.

Conjecture 3.1. Let f ∈ Sym 2d V be generic of SOS-rank k ≤ n and let N = n+d d . Then,

SOS k (f ) ∼ = {A ∈ C k ⊗ C N | A t A = W, xW x t = f } = {A ∈ C k ⊗ C N | A t A = W 0 + C 0 } = O(k).
Of course, if this intersection consists of more than a single point, one would arrive at exactly the number of copies of O(k) such that SOS k (f ) is isomorphic.

Consider a tuple (f 1 , . . . , f k ) ∈ SOS k (f ), we denote the tangent space of SOS k (f ) at this point by TSOS k (f ) (f1,...,f k ) . Recall that if we consider an orthogonal matrix O ∈ O(k) and A f ∈ M k×N , then the rows of A f O are polynomials giving a k-SOS decomposition of f . We are interested in understanding the local behavior of this variety. More specifically, we want to show that the tangent space TSOS k (f ) (f1,...,f k ) has dimension equal to the dimension of O(k). This means that locally, the variety SOS k (f ) is exactly equal to O(k). In order to do that, we can show that the only syzygies of a vector (f 1 , . . . , f k ) ∈ SOS k (f ) are given by the Koszul syzygies. In the next paragraphs, we further explain the concept of Koszul syzygies and how they are related to the tangent space of SOS k (f ).

Let A f be the matrix whose rows are the coefficients of f 1 , . . . , f k . Observe that the map

φ : A → xA t Ax t -f
gives SOS k (f ) as the fiber at zero. Therefore, the tangent space TSOS k (f ) (f1,...,f k ) is the space generated by the nullity of the derivative of φ at the point (f 1 , . . . , f k ). This equivalent to saying that

x(A t f V + V t A f )x t = 0 (4)
where V ∈ M k×N . Notice that equation ( 4) is trivially satisfied when A t f V is a skew-symmetric matrix. A syzygy satisfying this equation is a Koszul syzygy of (f 1 , . . . , f k ). If we have that the Koszul syzygies are the only syzygies of the point (f 1 , . . . , f k ), we obtain that they span the tangent space at this point. In such case, the tangent space has dimension equal to the dimension of O(k).

A more geometric and intuitive explanation can be described by looking at the usual set of coordinates instead of matrices. We may see SOS k (f ) as the nullity of the map

ϕ : (h 1 , . . . , h k ) → k i=1 h 2 i -f.
The tangent space TSOS k (f ) (f1,...,f k ) is computed once again as the space generated by the nullity of the derivative of the expression

k i=1 h 2 i -f at the point (f 1 , . . . , f k ), that is ϕ (f 1 , . . . , f k ) = 0.
This means that the tangent space is generated by

k i=1 f i g i = 0.
The vanishing of this expression by considering tuples (g 1 , . . . , g k ) such that we have pairs i = j with g i = f j and g j = -f i is a Koszul syzygy of the vector (f 1 , . . . , f k ). Observe that this corresponds exactly to the matrix A t f V being skew-symmetric, where V is the matrix that has g i as the ith-row.

Proposition 3.4. The only syzygies of the vector (x d 0 , . . . , x d k ) are the Koszul syzygies. Proof. Let A = I 0 be a matrix as in equation (2) giving a SOS decomposition of f

= x 2d 0 +• • •+x 2d k in a basis {x d 0 , . . . , x d n , . . . }, where I is the k × k-identity matrix. Consider V = v ij a k × N -matrix, then ∂ϕ(A) = V t A+A t V is the derivative of ϕ. The statement is equivalent to show that x∂ϕ(A)x t = 0 if and only if V t A is skew-symmetric.
In this basis,

W = V t A + A t V = v ij + v ji , and
xW x t = k i=1 k j=1 (v ij + v ji )x d j x d i = 0,
since each monomial coefficient is equal to zero we obtain 2(v ij + v ji ) = 0 as desired.

The importance of this result is that it guarantees that at the point (x d 0 , . . . , x d k ) the tangent space to SOS k (x 2d 0 + • • • + x 2d k ) has dimension equal to the number of Koszul syzygies, since they span the null space of ϕ (x d 0 , . . . , x d k ). Moreover, this dimension is exactly equal to the dimension of the tangent space of O(k). This implies that locally at the point (x d 0 , . . . , x d k ), the variety SOS k (f ) is equal to the subvariety O(k) ⊂ SOS k (f ). We wish to extend this result to every point (f 1 , . . . , f k ) ∈ SOS k (f ). We obtain that this can be extended to a vector (f 1 , . . . , f k ) by means of semi-continuity. Indeed, consider the kernel K of the map

O PV (-d) k [f1...f k ] -----→ O PV
defined by the vector (f 1 , . . . , f k ), where O PV is the sheaf defining PV as a scheme (PV, O P V ). The minimal resolution of the kernel, when there are only Koszul syzygies, start with

. . . → O PV (-2d) ( k 2 ) → K → 0
By Proposition 2.8, the Betti numbers β p,p+q of the minimal resolution of K correspond to the rank of Tor S p (K, k) p+q , this is the component of degree p + q of T or S p (K, k). Since we can correlate the Betti numbers with cohomology dimensions using Proposition 2.8, we have by Theorem 2.9 that for a local deformation of K, the Betti numbers satisfy semi-continuity. Moreover, since we know that for any other point (f 1 , . . . , f k ) will have at least the Koszul syzygies, this implies that it will have only them.

Corollary 3.5. Suppose that k ≤ n and f ∈ SOS k is general. Let (f 1 , . . . , f k ) be a vector in (Sym d V ) ×k giving the decomposition as k sum of squares of a polynomial f . Then the only syzygies of (f 1 , . . . , f k ) are the Koszul ones.

Corollary 3.6. Suppose that k ≤ n and f ∈ SOS k is general. We have an isomorphism SOS k (f ) ∼ = O(k) p , for some p ∈ Z + . Note that this does not depend on the degree of f . In particular

deg SOS k (f ) ≥ deg O(k) which is computed in Table 1, in fact deg SOS k (f ) ≡ 0 mod deg O(k).
We notice that the diagram 3 can have its conclusion interpreted in a different manner. Instead of considering W 0 a maximal rank matrix, one may consider a fixed matrix A 0 defining f , and let

SOS k (f ) = {B T B + C 0 | rank(B T B + C 0 ) = k, B T B = A T 0 A 0 and C 0 ∈ C}.
Notice that such interpretation means that adding C 0 = 0 is equivalent to changing the O(k) component of SOS k (f ). Thus, if there exists no other matrix C 0 besides 0 such that rank(A T 0 A 0 + C 0 ) = k, it implies that there exists only one component.

In the next pages we explore this equivalent problem and compare the dimensions of symmetric matrices of rank k and C. Although a proof that the only translation by C preserving the rank is 0 is not obtained, by a comparison of dimensions we get a clear indicator that we should not expect other solutions.

Let S N k be the variety of symmetric matrices of size N = n+d d of rank at most k. Then, for some fixed W ∈ S N k , consider the variety

(S + W ) N k = {B | B + W ∈ S N k }.
Note that this is indeed a variety as it is defined by the minors of the matrix B + W and moreover, for all M ∈ (S + W ) N k , we have that M -W ∈ S N k . Hence, we can consider this variety a translation of S N k by the matrix W .

(S + W ) N k = S N k -W. Recall the variety C = {C 0 ∈ S N | x T C 0 x = 0}
and note that the following statement holds:

For a generic W, (S + W ) N k ∩ C = 0 ⇐⇒ For a generic f, deg SOS k (f ) = deg O(k).
Firstly, note that since W is symmetric of rank k, there exists a decomposition of the form W = A T A where A ∈ M k×N . Then, since every symmetric matrix of size N gives a polynomial, through a moment vector x, we obtain a decomposition of x T W x as a sum of k squares as x T A T Ax. Then, as is discussed above, we would obtain equality for Corollary 3.6.

From the translation argument above, we obtain the following equivalences,

(S + W ) N k ∩ C = 0 ⇐⇒ (S N k -W ) ∩ C = 0 ⇐⇒ S N k ∩ (C + W ) = W.
The equations defining C are not general. Each equation specifies that a particular coefficient in the expansion of x t Bx be zero. Hence, no coefficients of a general f are zero, we have that a generic W is not contained in the hyperplanes defined by any of the n+2d

= (2N + 1 -k)k 2 .
Then, recall that

Sym 2 (Sym d V ) = Sym 2d V ⊕ C.
Thus,

codim C = dim Sym 2d V = n + 2d 2d .
The following lemma, through a dimension count, gives further support for Conjecture 3.1.

Lemma 3.7. Let k ≤ n. Then, for all n, d ≥ 1, dim S/I k < codim C.

Proof. Firstly, note that dim S/I k is maximal when k = N = n+d d and that the dimension decreases monotonically as k decreases. However, since we restrict to k ≤ n, it suffices to show that dim S/I n < codim C. Now, suppose that d = 1. Then,

dim S/I n -codim C = (2(n + 1) + 1 -n)n 2 - (n + 2)(n + 1) 2 = n(n + 3) -(n + 1)(n + 2) 2 = -1. Next, consider d ≥ 2. Note that for all n ≥ 1, (2 n+d d + 1 -n)n 2 ≤ n n + d d .
Hence, it suffices to prove that for all d ≥ 2,

n + 2d 2d > n n + d d .
We proceed by induction on d. In the base case d = 2 we have,

n + 4 4 -n n + 2 2 = (n + 4)(n + 3)(n + 2)(n + 1) 4! - (n + 2)(n + 1)n 2! = (n + 1)(n + 2) 4! (n 2 -5n + 12).
It is easy to see that the polynomial n 2 -5n + 12 is positive for all n and so the base case holds. Now, assume for some fixed d ≥ 2 that n+2d 2d > n n+d d and consider

n + 2d + 2 2d + 2 -n n + d + 1 d + 1 = (n + 2d + 2)(n + 2d + 1) (2d + 2)(2d + 1) n + 2d 2d - n + d + 1 d + 1 n n + d d = n n + d d (n + 2d + 2)(n + 2d + 1) (2d + 2)(2d + 1) n+2d 2d n n+d d - n + d + 1 d + 1 > n n + d d (n + 2d + 2)(n + 2d + 1) (2d + 2)(2d + 1) - n + d + 1 d + 1 = n n + d d 1 + n 2d + 2 1 + n 2d + 1 -1 + n d + 1 > n n + d d 1 + n 2d + 2 2 -1 + n d + 1 > n n + d d 1 + 2n 2d + 2 -1 + n d + 1 = 0.
Thus, by induction, codim C -dim S/I k > 0.

We finish this section by proving that Conjecture 3.1 holds for k = 2.

Theorem 1.4. Let f ∈ SOS 2 be a generic polynomial that is the sum of two squares. Then, SOS 2 (f ) has two irreducible components isomorphic to SO(2).

Proof. Let f ∈ Sym 2d V be a general polynomial such that f = g 2 + h 2 = (g + ih)(g -ih). Since this factorization is unique (UFD), we have for any other g , h such that f = g 2 + h 2 , then λ(g + ih ) = g + ih and λ -1 (g -ih ) = g -ih, or λ(g + ih ) = g -ih and λ -1 (g -ih ) = g + ih.

Consider the first set of conditions, then g = g λ+λ -1

2

+ h i(λ-λ -1 ) 2 and h = g λ-λ -1 2i + h λ+λ -1 2 . Thus g h = λ+λ -1 2 i(λ-λ -1 ) 2 λ-λ -1 2i λ+λ -1 2 A g h .
Since det(A) = 1 and AA t = I, this corresponds to one component of O(2). Then, the last copy of SO(2) is obtained from the other two conditions.

The degree of the variety of the sum of two squares

Let V be a complex vector space of dimension n + 1 and let d ≥ 0 be an integer. Let U = Sym d V and π C be the projection of Sym

2 U = Sym 2d V ⊕ C centered at C, that is π C : Sym 2d V ⊕ C → Sym 2d V. Notice that whenever Σ k (ν 2 (U )) ∩ C = 0, π C | Σ k (ν2(U )) is a well-defined morphism. In such a case, assuming π C | Σ k (ν2(U )) is an isomorphism, this means that deg(SOS k ) = deg Σ k (ν 2 (U )) .
Theorem 4.1. Following the previous notation we have that

Σ 1 (ν 2 (PU )) ∩ C = ∅ and Σ 2 (ν 2 (PU )) ∩ C = ∅.
Proof. It is known from the Borel-Weil Theorem, see [START_REF] Weyman | Cohomology of Vector Bundles and Syzygies[END_REF], that whenever X = G/P ⊂ P(V λ ), where G is an algebraic group and P ⊂ G a parabolic group, then H 0 (X, P(V λ (k))) = V kλ [START_REF] Serre | Représentations linéaires et espaces homogenes kählériens des groupes de lie compacts[END_REF]. If we consider X = ν d (PV ) ⊂ PU and Sym 2d V = V 2d , we have the short exact sequence

0 → I X → O PU → O X → 0.
Twisting it by O PU (2) and taking the long exact sequence of cohomologies we obtain

0 → H 0 (I X (2)) → H 0 (O PU (2)) → H 0 (X, O X (2)) → 0.
Notice that the last map is a surjection since H 0 (X, O X (2)) = V 2d that is irreducible.

We remark the follow identifications: H 0 (I X (2)) is given by the quadric forms on the ideal sheaf of X, that is, the quadric forms that belong to

C. H 0 (O PU (2)) = Sym 2 (U ) = Sym 2 (Sym d V ) and H 0 (X, O X (2)) = Sym 2d V .
Assume for the purpose of contradiction that Σ 1 (ν 2 (PU )) ∩ C = ∅ and Σ 2 (ν 2 (PU )) ∩ C = ∅, this means that there exists polynomials f, g ∈ C of respective ranks 1 and 2 in P Sym 2 U such that f, g ∈ H 0 (I X (2)). This implies that X is contained in the hyperplane determined by f = l 2 and in the union of hyperplanes determined by g = l 2 0 + l 2 1 = (l 0 + il 1 )(l 0 -il 1 ), where l, l 0 , l 1 are linear forms in PU . However X = ν 2 (PU ) is not contained in any hyperplane, thus the intersection must be empty.

Lemma 4.2. The map π C is injective in ν 2 (PU ).

Proof. Let x, y be elements both in ν 2 (PU ). The map is given by x → x, C ∩ Sym 2d V.

Thus, the equation π C (x) = π C (y) implies that x, C = y, C. This means that there exists λ ∈ C and c ∈ C such that x = λy + c. Therefore, x -λy = c ∈ C which is a contradiction since x -λy ∈ Σ 2 (ν 2 (U )).

Lemma 4.3. The projection π C : Sym 2 (Sym d V ) → Sym 2d V restricted to the second secant variety of the Veronese variety Σ 2 (ν 2 (Sym d V )) is injective.

Proof. Consider the projection

Sym d V × Sym d V × Sym 2 (Sym d V ) → Sym 2 (Sym d V ).
Let Ab 2 (ν 2 (Sym d V )) = {(α, β, g)|α 2 + β 2 = g} be the abstract Veronese variety that under the projection is mapped to Σ 2 (ν 2 (Sym d V )). Notice that the fiber of this projection on a point g is O(2) by Lemma 4.2.

We may consider a similar projection

Sym d V × Sym d V × Sym 2d V → Sym 2d V.
We may define X = {(α, β, f )|α 2 + β 2 = f } in the same fashion as before. Under this projection we have that X is mapped to SOS 2 and the fiber on a point f is SOS 2 (f ) = O(2) by Lemma 1.4. Notice that the map Sym 2 (Sym d V ) → Sym 2d V that corresponds to the change of coordinates w 1 = x d 0 , . . . , w N = x d n is injective when restricted to Σ 2 (ν 2 (Sym d V )) and so is the induced linear map from Ab 2 to X.

Joining those maps into a diagram we obtain:

Ab 2 X Σ 2 (ν 2 (Sym d V )) SOS 2 ψ ϕ ξ ζ
From the previous remarks, ϕ is an one-to-one map and the fibers of ψ and ξ are both equal to O(2).

Since the diagram commutes, we also obtain that ζ is a one-to-one map. .

We notice that in the case of n = 2 and d = 2 Theorem 4.1 is sharp in the sense that for the 3-secant variety of ν 2 (PU ) the intersection with C is non-empty. Indeed, one can find by computation that the intersection of Σ 1 (ν 2 (PU )) and Σ 2 (ν 2 (PU )) with C are empty. Thus, deg(SOS 1 ) = 32 and deg(SOS 2 ) = 126 as expected. However, for SOS 3 the intersection has codimension 3 in P 5 = PU . When the intersection is non-empty, the degree of Σ k (ν 2 (PU )) is still an upper bound for the degree of SOS k .

2d

  equations defining C. Let N = n+d d and let S be the polynomial ring C[x ij |1 ≤ i, j ≤ N ]. We set x ij = x ji and consider X = (x ij ) 1≤i,j≤N to be an N × N variable symmetric matrix. For 1 ≤ k ≤ N -1, we denote by I k the ideal generated by the k + 1 minors of X. It is known that S/I k is a Cohen-Macaulay normal domain with dimension dim S/I k

Theorem 1. 6 .

 6 Let N = dim Sym d V = n+d d . The degrees of the varieties of squares and of sum of two squares in P(Sym 2d V ) are given bydeg(SOS 1 ) = 2 N -1 , deg(SOS 2 ) = Since Σ k (ν 2 (PU )) ∩ C = ∅ and π C | Σ k (ν2(PU )) is injective for k = 1, 2, it follows deg(SOS j ) = deg(Σ j (ν 2 (PU ))).A classical result by Segre[START_REF] Harris | On symmetric and skew-symmetric determinantal varieties[END_REF] states that for any j ≤ N deg(Σ j (ν 2 (PU ))) =

Table 1 :

 1 Degree of SOS k (f ) for n ≥ k. See formula 1 for the degree of O(k).

		4	4	2
	3	16	16	8
	4	80	80	40
	5	768	768	384
	6	9356	9356	4768
	7	-	233232	111616
	8	-	6867200	3433600
	9	-	393936896	196968448

Acknowledgements

This work has been supported by European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions, grant agreement 813211 (POEMA).