Matrix-F5 algorithms and tropical Gr\"obner bases computation

Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr\"obner bases taking into account the valuation of $K$. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a $p$-adic fields.We design a strategy to compute such tropical Gr\"obner bases by adapting the Matrix-F5 algorithm. Two variants of the Matrix-F5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster.Our study is performed both over any exact field with valuation and some inexact fields like $\mathbb{Q}\_p$ or $\mathbb{F}\_q \llbracket t \rrbracket.$ In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Gr\"obner bases when the valuation is non-trivial. Numerical examples are provided

Towards Toric Absolute Factorization

International audienceThis article gives an algorithm to recover the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry

Approche effective des résidus algébriques

Dans ce rapport, nous nous intéressons aux propriétés du dual des polynômes et leurs connections avec la théorie algébrique des résidus. En particulier, nous voulons éclaircir le rôle que jouent les Bézoutiens dans la théorie des résidus et dans la résolution de systèmes polynomiaux. Nous rappelons donc les propriétés du dual des polynômes et des systèmes inverses. Nous montrons dans le cas local (point isolé) comment ces systèmes inverses peuvent être calculés par intégration. Puis nous détaillons les travaux de G. Scheja et U. Storch \cite{ScSt75}, et E. Kunz \cite{Kun86}, où les fondements de la théorie algébrique des résidus sont établis. Nous étudions ensuite le comportement du résidu sous l'action de certaines transformations polynomiales, et le lien avec les bases de Gröbner. Nous donnons enfin des preuves élémentaires de résultats classiques tels que le théorème de Macaulay, la formule d'Euler-Jacobi, le théorème de Bézout, la formule de Weil, et des algorithmes pour le calcul du résidu, dans certains cas, en utilisant les bases de Gröbner ou en résolvant des systèmes linéaires de tailles raisonnables

On mixed polynomials of bidegree (n, 1)

Specifying the bidegrees (n, m) of mixed polynomials P (z, ¯ z) of the single complex variable z, with complex coefficients, allows to investigate interesting roots structures and counting; intermediate between complex and real algebra. Multivariate mixed polynomials appeared in recent papers dealing with Milnor fibrations, but in this paper we focus on the univariate case and m = 1, which is closely related to the important subject of harmonic maps. Here we adapt, to this setting, two algorithms of computer algebra: Vandermonde interpolation and a bissection-exclusion method for root isolation. Implemented in Maple, they are used to explore some interesting classes of examples

Intersection and self-intersection of surfaces by means of Bezoutian matrices

International audienceThe computation of intersection and self-intersection loci of parameterized surfaces is an important task in Computer Aided Geometric Design. Computer algebra tools need to be developed further for computing their implicit equations. We address these problems via four resultants with separated variables. Two of them are specializations of more general ones and the others are determinantal. We give a rigorous study in these cases and provide new and useful formulas via adapted computations of Bezoutians

A computational study of ruled surfaces

International audienceWe study rational ruled surfaces and $\mu$-bases which were recently considered in a series of articles by Chen and coworkers. We give short and conceptual proofs with geometric insights and efficient algorithms. In particular, we provide a method to reparameterize an improper parameterization and we also briefly explain how to deal with approximate input data. Finally we provide an algorithmic description of self-intersection loci

Using projection operators in computer aided geometric design

We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention different formulations of resultants, including the projective resultant, the toric resultant, and the residual resultant. In the second part we illustrate these tools, and other projection operators, on typical problems of CAGD such as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface

Identification of phytochemicals of Phoenix dactylifera L. Cv Ajwa with UHPLC-ESI-QTOF-MS/MS

Date palm (Phoenix dactylifera) fruit is an important food in the Middle East. Some date palm cultivars like the Ajwa is revered for its health-promoting properties. Phytochemicals like phenolic compounds are in part responsible for such health benefits. However, detailed information on these compounds is lacking. The present work aimed to investigate phytochemical components of Ajwa cultivar using UHPLC-ESI-QTOF-MS/MS in three different extraction solvents (aqueous, methanol–water, acetone–water) and three different extraction durations (2, 5 and 24 h). The proposed method provided tentative identification of 169 bioactive compounds out of which 44 (polyphenols and other phytochemical compounds) were successfully identified from three different extracts and three different extraction durations. Twenty-one compounds never previously reported in the Ajwa cultivar were identified. Aqueous-based extraction solvent and 24-h extraction duration yielded most phytochemical compounds

Mortality and pulmonary complications in patients undergoing surgery with perioperative SARS-CoV-2 infection: an international cohort study

Background: The impact of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) on postoperative recovery needs to be understood to inform clinical decision making during and after the COVID-19 pandemic. This study reports 30-day mortality and pulmonary complication rates in patients with perioperative SARS-CoV-2 infection. Methods: This international, multicentre, cohort study at 235 hospitals in 24 countries included all patients undergoing surgery who had SARS-CoV-2 infection confirmed within 7 days before or 30 days after surgery. The primary outcome measure was 30-day postoperative mortality and was assessed in all enrolled patients. The main secondary outcome measure was pulmonary complications, defined as pneumonia, acute respiratory distress syndrome, or unexpected postoperative ventilation. Findings: This analysis includes 1128 patients who had surgery between Jan 1 and March 31, 2020, of whom 835 (74·0%) had emergency surgery and 280 (24·8%) had elective surgery. SARS-CoV-2 infection was confirmed preoperatively in 294 (26·1%) patients. 30-day mortality was 23·8% (268 of 1128). Pulmonary complications occurred in 577 (51·2%) of 1128 patients; 30-day mortality in these patients was 38·0% (219 of 577), accounting for 81·7% (219 of 268) of all deaths. In adjusted analyses, 30-day mortality was associated with male sex (odds ratio 1·75 [95% CI 1·28–2·40], p\textless0·0001), age 70 years or older versus younger than 70 years (2·30 [1·65–3·22], p\textless0·0001), American Society of Anesthesiologists grades 3–5 versus grades 1–2 (2·35 [1·57–3·53], p\textless0·0001), malignant versus benign or obstetric diagnosis (1·55 [1·01–2·39], p=0·046), emergency versus elective surgery (1·67 [1·06–2·63], p=0·026), and major versus minor surgery (1·52 [1·01–2·31], p=0·047). Interpretation: Postoperative pulmonary complications occur in half of patients with perioperative SARS-CoV-2 infection and are associated with high mortality. Thresholds for surgery during the COVID-19 pandemic should be higher than during normal practice, particularly in men aged 70 years and older. Consideration should be given for postponing non-urgent procedures and promoting non-operative treatment to delay or avoid the need for surgery. Funding: National Institute for Health Research (NIHR), Association of Coloproctology of Great Britain and Ireland, Bowel and Cancer Research, Bowel Disease Research Foundation, Association of Upper Gastrointestinal Surgeons, British Association of Surgical Oncology, British Gynaecological Cancer Society, European Society of Coloproctology, NIHR Academy, Sarcoma UK, Vascular Society for Great Britain and Ireland, and Yorkshire Cancer Research

Outcomes from elective colorectal cancer surgery during the SARS-CoV-2 pandemic

This study aimed to describe the change in surgical practice and the impact of SARS-CoV-2 on mortality after surgical resection of colorectal cancer during the initial phases of the SARS-CoV-2 pandemic