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In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when $\operatorname{Spec}(R)$ is; this is the degree-zero case of our result on polynomial functors., Comment: 35 pages, 1 figure

The CSSI 2019 workshop was held on October 28-29, 2019, in Austin, Texas. The main objectives of this workshop were to (1) understand the impact of the CSSI program on the community over the last 9 years, (2) engage workshop participants in identifying gaps and opportunities in the current CSSI landscape, (3) gather ideas on the cyberinfrastructure needs and expectations of the community with respect to the CSSI program, and (4) prepare a report summarizing the feedback gathered from the community that can inform the future solicitations of the CSSI program. The workshop brought together different stakeholders interested in provisioning sustainable cyberinfrastructure that can power discoveries impacting the various fields of science and technology and maintaining the nation's competitiveness in the areas such as scientific software, HPC, networking, cybersecurity, and data/information science. The workshop served as a venue for gathering the community-feedback on the current state of the CSSI program and its future directions., Comment: This report was submitted in April 2020 to the National Science Foundation (NSF)

We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian., Comment: 16 pages

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fr\"oberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree $7$ and in degree $9$., Comment: 19 pages, final revision

The inner regions of the discs of high-mass young stellar objects (HMYSOs) are still poorly known due to the small angular scales and the high visual extinction involved. We deploy near-infrared (NIR) spectro-interferometry to probe the inner gaseous disc in HMYSOs and investigate the origin and physical characteristics of the CO bandhead emission (2.3-2.4 $\mu$m). We present the first GRAVITY/VLTI observations at high spectral (R=4000) and spatial (mas) resolution of the CO overtone transitions in NGC 2024 IRS2. The continuum emission is resolved in all baselines and is slightly asymmetric, displaying small closure phases ($\leq$8$^{\circ}$). Our best ellipsoid model provides a disc inclination of 34$^{\circ}$$\pm$1$^{\circ}$, a disc major axis position angle of 166$^{\circ}$$\pm$1$^{\circ}$, and a disc diameter of 3.99$\pm$0.09 mas (or 1.69$\pm$0.04 au, at a distance of 423 pc). The small closure phase signals in the continuum are modelled with a skewed rim, originating from a pure inclination effect. For the first time, our observations spatially and spectrally resolve the first four CO bandheads. Changes in visibility, as well as differential and closure phases across the bandheads are detected. Both the size and geometry of the CO-emitting region are determined by fitting a bidimensional Gaussian to the continuum-compensated CO bandhead visibilities. The CO-emitting region has a diameter of 2.74$\pm^{0.08}_{0.07}$ mas (1.16$\pm$0.03 au), and is located in the inner gaseous disc, well within the dusty rim, with inclination and $PA$ matching the dusty disc geometry, which indicates that both dusty and gaseous discs are coplanar. Physical and dynamical gas conditions are inferred by modelling the CO spectrum. Finally, we derive a direct measurement of the stellar mass of $M_*\sim$14.7$^{+2}_{-3.6}$ M$_{\odot}$ by combining our interferometric and spectral modelling results., Comment: Accepted for publication in A&A letters

Using MUSE AO data, we probe the ISM in the local spiral galaxy NGC 7793 at a spatial resolution of $\sim$ 10 pc. We identify HII regions and compile a catalogue of supernova remnants (SNRs), planetary nebulae (PNe) and Wolf Rayet stars (WR). We compute electron densities and temperatures from the [SII]6716/6731 and [SIII]6312/9069 line ratios. We study the properties of the ionised gas through BPT diagrams combined with gas velocity dispersion. We spectroscopically confirm 2 previously detected WR stars and a SNR and report the discovery of 7 WR, 1 SNR, and 2 PNe. The diffuse ionized gas (DIG) fraction is between $\sim$ 27 and 42% depending on the method used to define the HII region boundaries. In agreement with previous studies, we find that the DIG exhibits enhanced [SII]/H$\alpha$ and [NII]/H$\alpha$ ratios and a median temperature that is $\sim$ 3000 K higher than in HII regions. We also observe an apparent inverse correlation between temperature and H$\alpha$ surface brightness. Overall, the observed [SII]6716/6731 ratio is consistent within 1$\sigma$ with $n_e$ < 30 cm$^{-3}$, with an almost identical distribution for the DIG and HII regions. The velocity dispersion of the ionised gas indicates that the DIG has a higher degree of turbulence than the HII regions. Comparison with photoionisation and shock models reveals that the diffuse component can only partially be explained via shocks and that it is most likely consistent with photons leaking from density bounded HII regions or with radiation from evolved field stars. [abridged], Comment: Accepted for publication in A&A. 24 pages, 19 figures. Please see journal article for high resolution figures

The process of massive star ($M\geq8~M_\odot$) formation is still poorly understood. Observations of massive young stellar objects (MYSOs) are challenging due to their rarity, short formation timescale, large distances, and high circumstellar extinction. Here, we present the results of a spectroscopic analysis of a population of MYSOs in the Large Magellanic Cloud (LMC). We took advantage of the spectral resolution and wavelength coverage of X-shooter (300-2500 nm), mounted on the European Southern Observatory Very Large Telescope, to detect characteristic spectral features in a dozen MYSO candidates near 30 Doradus, the largest starburst region in the Local Group hosting the most massive stars known. The X-shooter spectra are strongly contaminated by nebular emission. We used a scaling method to subtract the nebular contamination from our objects. We detect H$\alpha,\beta$, [O I] 630.0 nm, Ca II infrared triplet, [Fe II] 1643.5 nm, fluorescent Fe II 1687.8 nm, H$_2$ 2121.8 nm, Br$\gamma$, and CO bandhead emission in the spectra of multiple candidates. This leads to the spectroscopic confirmation of 10 candidates as bona fide MYSOs. We compare our observations with photometric observations from the literature and find all MYSOs to have a strong near-infrared excess. We compute lower limits to the brightness and luminosity of the MYSO candidates, confirming the near-infrared excess and the massive nature of the objects. No clear correlation is seen between the Br$\gamma$ luminosity and metallicity. Combining our sample with other LMC samples results in a combined detection rate of disk features such as fluorescent Fe II and CO bandheads which is consistent with the Galactic rate (40\%). Most of our MYSOs show outflow features., Comment: Accepted to A&A, 26 pages, 19 figures, 4 tables

Context: The identification and characterisation of populations of young massive stars in (giant) HII regions provides important constraints on i) the formation process of massive stars and their early feedback on the environment, and ii) the initial conditions for population synthesis models predicting the evolution of ensembles of stars. Aims: We identify and characterise the stellar populations of the following young giant HII regions: M8, G333.6-0.2, and NGC6357. Methods: We acquired H- and K-band spectra of around 200 stars using The K-band KMOS on the ESO Very Large Telescope. The targets for M8 and NGC6357 were selected from the MYStIX project, which combines X-ray observations with near-infrared and mid-infrared data. For G333.6-0.2, the sample selection is based on the near-infrared colours combined with X-ray data. We introduce an automatic spectral classification method in order to obtain temperatures and luminosities for the observed stars. We analyse the stellar populations using their photometric, astrometric, and spectroscopic properties and compared the position of the stars in the Hertzprung-Russell diagram with stellar evolution models to constrain their ages and mass ranges. Results: We confirm the presence of candidate ionising sources in the three regions and report new ones, including the first spectroscopically identified O stars in G333.6-0.2. In M8 and NGC6357, two populations are identified: i) OB main-sequence stars ($M > 5~\rm{M_{\odot}}$) and ii) pre-main sequence stars ($M\approx0.5-5~\rm{M_{\odot}}$). The ages of the clusters are $\sim$1-3~Myr, $< 3$~Myr, and $\sim$0.5-3~Myr for M8, G333.6-0.2, and NGC6357, respectively. We show that MYStIX selected targets have $>$ 90\% probability of being members of the HII region, whereas a selection based on near infrared (NIR) colours leads to a membership probability of only $\sim$70\%., Comment: Accepted for publication in A&A. The arXiv version includes appendix C, which is an online only figure in A&A

We find a semi-algebraic description of the Minkowski sum $\mathcal{A}_{3,n}$ of $n$ copies of the bounded twisted cubic $\{(t,t^2,t^3)\mid -1\leq t\leq 1\}$ for each integer $n\geq3$. These descriptions provide efficient membership tests for the sets $\mathcal{A}_{3,n}$. These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was formulated by Michael Rubinstein and Peter Sarnak in order to study problems related to $L$-functions and their zeros., Comment: 19 pages, 3 figures, major revision

We have used the capability of the MUSE instrument to explore the impact of stellar feedback at large scales in Haro 11, a galaxy under extreme starburst condition and one of the first galaxies where Lyman continuum (LyC) has been detected. Using Ha, [OIII] and [OI] emission lines from deep MUSE observations, we have constructed a sequence of velocity-dependent maps of the Ha emission, the state of the ionised gas and a tracer of fast shocks. These allowed us to investigate the ionisation structure of the galaxy in 50 kms^2 bins over a velocity range of -400 to 350 kms. The ionised gas in Haro 11 is assembled by a rich arrangement of structures, such as superbubbles, filaments, arcs and galactic ionised channels, whose appearances change drastically with velocity. The central star forming knots and the star forming dusty arm are the main engines that power the strong mechanical feedback in this galaxy, although with different impact on the ionisation structure. Haro 11 appears to leak LyC radiation in many directions. We found evidence of a kpc-scale fragmented superbubble, that may have cleared galactic-scale channels in the ISM. Additionally, the southwestern hemisphere is highly ionised in all velocities, hinting at a density bound scenario. A compact kpc-scale structure of lowly ionised gas coincides with the diffuse Lya emission and the presence of fast shocks. Finally, we find evidence that a significant fraction of the ionised gas mass may escape the gravitational potential of the galaxy., Comment: Pubisched version. Ionisation values were corrected after discovering a bug in a code used to generate the maps

X-ray radiation from accreting compact objects is an important part of stellar feedback. The metal-poor galaxy ESO 338-4 has experienced vigorous starburst during the last 40 Myr and contains some of the most massive super star clusters in the nearby Universe. Given its starburst age and its star-formation rate, ESO 338-4 is one of the most efficient nearby manufactures of neutron stars and black holes, hence providing an excellent laboratory for feedback studies. We compared X-ray images and spectra obtained by XMM-Newton and Chandra telescopes with integral field spectroscopic VLT MUSE observations in the optical to constrain the nature of strong X-ray emitters. X-ray observations uncover three ultraluminous X-ray sources (ULXs) in ESO 338-4. The brightest among them, ESO 338~X-1, has X-ray luminosity in excess of 10^{40} erg/s. We speculate that ESO 338-4 is powered by accretion on an intermediate-mass (~300Msun) black hole. We show that X-ray radiation from ULXs and hot superbubbles strongly contributes to HeII ionization and general stellar feedback in this template starburst galaxy., Comment: A&A, in press

The interplay between the formation of stars, stellar feedback and cloud properties strongly influences the star formation history of giant molecular clouds. The formation of massive stars leads to a variety of stellar clusters, ranging from low stellar density OB associations to dense, gravitationally bound starburst clusters. We aimed at identifying the massive stellar content and reconstructing the star formation history of the W51 giant molecular cloud. We performed near-infrared imaging and K-band spectroscopy of the massive stars in W51. We analyzed the stellar populations using colour-magnitude and colour-colour diagrams and compared the properties of the spectroscopically identified stars with stellar evolution models. We derived the ages of the different sub-clusters in W51 and, based on our spectroscopy derived an age for W51 of 3 Myrs or less. The age of the P Cygni star LS1 and the presence of two still forming proto-clusters suggests that the star formation history of W51 is more complex than a single burst. We did not find evidence for triggered star formation and we concluded that the star formation in W51 is multi seeded. We finally concluded that W51 is a OB association where different sub-clusters form over a time span of at least 3-5 Myrs., Comment: 21 Pages, 9 figures, accepted for publication by Astronomy and Astrophysics

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances., Comment: 26 pages, added a discussion on the monic rank for reducible cones

Stellar feedback strongly affects the interstellar medium (ISM) of galaxies. Stellar feedback in the first galaxies likely plays a major role in enabling the escape of LyC photons, which contribute to the re-ionization of the Universe. Nearby starburst galaxies serve as local analogues allowing for a spatially resolved assessment of the feedback processes in these galaxies. We characterize the feedback effects from the star clusters in the local high-redshift analogue ESO 338-IG04 on the ISM and compare the results with the properties of the most massive clusters. We use high quality VLT/MUSE optical integral field data to derive the physical properties of the ISM such as ionization, density, shocks, and perform new fitting of the spectral energy distributions of the brightest clusters in ESO 338-IG04 from HST imaging. ESO 338-IG04 has a large ionized halo which we detect to a distance of 9 kpc. We identify 4 Wolf-Rayet (WR) clusters based on the blue and red WR bump. We follow previously identified ionization cones and find that the ionization of the halo increases with distance. Analysis of the galaxy kinematics shows two complex outflows driven by the numerous young clusters in the galaxy. We find a ring of shocked emission traced by an enhanced [OI]/H$\alpha$ ratio surrounding the starburst and at the end of the outflow. Finally we detect nitrogen enriched gas associated with the outflow, likely caused by the WR stars in the massive star clusters. Photo-ionization dominates the central starburst and sets the ionization structure of the entire halo, resulting in a density bounded halo, facilitating the escape of LyC photons. Outside the central starburst, shocks triggered by an expanding super bubble become important. The shocks at the end of the outflow suggest interaction between the hot outflowing material and the more quiescent halo gas., Comment: Accepted for publication in Astronomy and Astrophysics, 22 pages, 15 figures