Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polyomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials

The properties of ten O-type stars in the low-metallicity galaxies IC 1613, WLM and NGC 3109

Massive stars likely played an important role in the reionization of the Universe, and the formation of the first black holes. Massive stars in low-metallicity environments in the local Universe are reminiscent of their high redshift counterparts. In a previous paper, we reported on indications that the stellar winds of low-metallicity O stars may be stronger than predicted, which would challenge the current paradigm of massive star evolution. In this paper, we aim to extend our initial sample of six O stars in low-metallicity environments by four. We aim to derive their stellar and wind parameters, and compare these to radiation-driven wind theory and stellar evolution models. We have obtained intermediate-resolution VLT/X-Shooter spectra of our sample of stars. We derive the stellar parameters by fitting synthetic fastwind line profiles to the VLT/X-Shooter spectra using a genetic fitting algoritm. We compare our parameters to evolutionary tracks and obtain evolutionary masses and ages. We also investigate the effective temperature versus spectral type calibration for SMC and lower metallicities. Finally, we reassess the wind momentum versus luminosity diagram. The derived parameters of our target stars indicate stellar masses that reach values of up to 50 $M_{\odot}$. The wind strengths of our stars are, on average, stronger than predicted from radiation-driven wind theory and reminiscent of stars with an LMC metallicity. We discuss indications that the iron content of the host galaxies is higher than originally thought and is instead SMC-like. We find that the discrepancy with theory is lessened, but remains significant for this higher metallicity. This may imply that our current understanding of the wind properties of massive stars, both in the local universe as well as at cosmic distances, remains incomplete.Comment: Accepted for publication in Astronomy and Astrophysics. 10 pages, 8 figure

The VLT-FLAMES Tarantula Survey

We present a number of notable results from the VLT-FLAMES Tarantula Survey (VFTS), an ESO Large Program during which we obtained multi-epoch medium-resolution optical spectroscopy of a very large sample of over 800 massive stars in the 30 Doradus region of the Large Magellanic Cloud (LMC). This unprecedented data-set has enabled us to address some key questions regarding atmospheres and winds, as well as the evolution of (very) massive stars. Here we focus on O-type runaways, the width of the main sequence, and the mass-loss rates for (very) massive stars. We also provide indications for the presence of a top-heavy initial mass function (IMF) in 30 Dor.Comment: 7 Figures, 8 pages. Invited talk: IAUS 329: "The Lives and Death-Throes of Massive Stars

La habichuela.

Habichuel

The VLT-FLAMES Tarantula Survey XXI. Stellar spin rates of O-type spectroscopic binaries

The initial distribution of spin rates of massive stars is a fingerprint of their elusive formation process. It also sets a key initial condition for stellar evolution and is thus an important ingredient in stellar population synthesis. So far, most studies have focused on single stars. Most O stars are however found in multiple systems. By establishing the spin-rate distribution of a sizeable sample of O-type spectroscopic binaries and by comparing the distributions of binary sub-populations with one another as well as with that of presumed single stars in the same region, we aim to constrain the initial spin distribution of O stars in binaries, and to identify signatures of the physical mechanisms that affect the evolution of the massive stars spin rates. We use ground-based optical spectroscopy obtained in the framework of the VLT-FLAMES Tarantula Survey (VFTS) to establish the projected equatorial rotational velocities (\vrot) for components of 114 spectroscopic binaries in 30 Doradus. The \vrot\ values are derived from the full-width at half-maximum (FWHM) of a set of spectral lines, using a FWHM vs. \vrot\ calibration that we derive based on previous line analysis methods applied to single O-type stars in the VFTS sample. The overall \vrot\ distribution of the primary stars resembles that of single O-type stars in the VFTS, featuring a low-velocity peak (at \vrot < 200 kms) and a shoulder at intermediate velocities (200 < \vrot < 300 kms). The distributions of binaries and single stars however differ in two ways. First, the main peak at \vrot \sim100 kms is broader and slightly shifted toward higher spin rates in the binary distribution compared to that of the presumed-single stars. Second, the \vrot distribution of primaries lacks a significant population of stars spinning faster than 300 kms while such a population is clearly present in the single star sample.Comment: 16 pages, 16 figures, paper accepted in Astronomy & Astrophysic