Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page

Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is {\em polynomial} in the length of the input (given in straight--line program representation) and an adequately defined {\em geometric degree of the equation system}. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined {\em real (or complex) degree} of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.Comment: Late

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.Comment: Late

A Trial of Early Antiretrovirals and Isoniazid Preventive Therapy in Africa

BACKGROUND: In sub-Saharan Africa, the burden of human immunodeficiency virus (HIV)-associated tuberculosis is high. We conducted a trial with a 2-by-2 factorial design to assess the benefits of early antiretroviral therapy (ART), 6-month isoniazid preventive therapy (IPT), or both among HIV-infected adults with high CD4+ cell counts in Ivory Coast. METHODS: We included participants who had HIV type 1 infection and a CD4+ count of less than 800 cells per cubic millimeter and who met no criteria for starting ART according to World Health Organization (WHO) guidelines. Participants were randomly assigned to one of four treatment groups: deferred ART (ART initiation according to WHO criteria), deferred ART plus IPT, early ART (immediate ART initiation), or early ART plus IPT. The primary end point was a composite of diseases included in the case definition of the acquired immunodeficiency syndrome (AIDS), non-AIDS-defining cancer, non-AIDS-defining invasive bacterial disease, or death from any cause at 30 months. We used Cox proportional models to compare outcomes between the deferred-ART and early-ART strategies and between the IPT and no-IPT strategies. RESULTS: A total of 2056 patients (41% with a baseline CD4+ count of ≥500 cells per cubic millimeter) were followed for 4757 patient-years. A total of 204 primary end-point events were observed (3.8 events per 100 person-years; 95% confidence interval [CI], 3.3 to 4.4), including 68 in patients with a baseline CD4+ count of at least 500 cells per cubic millimeter (3.2 events per 100 person-years; 95% CI, 2.4 to 4.0). Tuberculosis and invasive bacterial diseases accounted for 42% and 27% of primary end-point events, respectively. The risk of death or severe HIV-related illness was lower with early ART than with deferred ART (adjusted hazard ratio, 0.56; 95% CI, 0.41 to 0.76; adjusted hazard ratio among patients with a baseline CD4+ count of ≥500 cells per cubic millimeter, 0.56; 95% CI, 0.33 to 0.94) and lower with IPT than with no IPT (adjusted hazard ratio, 0.65; 95% CI, 0.48 to 0.88; adjusted hazard ratio among patients with a baseline CD4+ count of ≥500 cells per cubic millimeter, 0.61; 95% CI, 0.36 to 1.01). The 30-month probability of grade 3 or 4 adverse events did not differ significantly among the strategies. CONCLUSIONS: In this African country, immediate ART and 6 months of IPT independently led to lower rates of severe illness than did deferred ART and no IPT, both overall and among patients with CD4+ counts of at least 500 cells per cubic millimeter. (Funded by the French National Agency for Research on AIDS and Viral Hepatitis; TEMPRANO ANRS 12136 ClinicalTrials.gov number, NCT00495651.)

Assessing the feasibility, acceptability, and fidelity of a tele-retinopathy-based intervention to encourage greater attendance to diabetic retinopathy screening in immigrants living with diabetes from China and African-Caribbean countries in Ottawa, Canada: a protocol

Background: Diabetic retinopathy is a leading cause of preventable blindness in Canada. Clinical guidelines recommend annual diabetic retinopathy screening for people living with diabetes to reduce the risk and progression of vision loss. However, many Canadians with diabetes do not attend screening. Screening rates are even lower in immigrants to Canada including people from China, Africa, and the Caribbean, and these groups are also at higher risk of developing diabetes complications. We aim to assess the feasibility, acceptability, and fidelity of a co-developed, linguistically and culturally tailored tele-retinopathy screening intervention for Mandarin-speaking immigrants from China and French-speaking immigrants from African-Caribbean countries living with diabetes in Ottawa, Canada, and identify how many from each population group attend screening during the pilot period. // Methods: We will work with our health system and patient partners to conduct a 6-month feasibility pilot of a tele-retinopathy screening intervention in a Community Health Centre in Ottawa. We anticipate recruiting 50–150 patients and 5–10 health care providers involved in delivering the intervention for the pilot. Acceptability will be assessed via a Theoretical Framework of Acceptability-informed survey with patients and health care providers. To assess feasibility, we will use a Theoretical Domains Framework-informed interview guide and to assess fidelity, and we will use a survey informed by the National Institutes of Health framework from the perspective of health care providers. We will also collect patient demographics (i.e., age, gender, ethnicity, health insurance status, and immigration information), screening outcomes (i.e., patients with retinopathy identified, patients requiring specialist care), patient costs, and other intervention-related variables such as preferred language. Survey data will be descriptively analyzed and qualitative data will undergo content analysis. // Discussion: This feasibility pilot study will capture how many people living with diabetes from each group attend the diabetic retinopathy screening, costs, and implementation processes for the tele-retinopathy screening intervention. The study will indicate the practicability and suitability of the intervention in increasing screening attendance in the target population groups. The study results will inform a patient-randomized trial, provide evidence to conduct an economic evaluation of the intervention, and optimize the community-based intervention

Effect of isoniazid preventive therapy on risk of death in west African, HIV-infected adults with high CD4 cell counts: long-term follow-up of the Temprano ANRS 12136 trial.

BACKGROUND: Temprano ANRS 12136 was a factorial 2 × 2 trial that assessed the benefits of early antiretroviral therapy (ART; ie, in patients who had not reached the CD4 cell count threshold used to recommend starting ART, as per the WHO guidelines that were the standard during the study period) and 6-month isoniazid preventive therapy (IPT) in HIV-infected adults in Côte d'Ivoire. Early ART and IPT were shown to independently reduce the risk of severe morbidity at 30 months. Here, we present the efficacy of IPT in reducing mortality from the long-term follow-up of Temprano. METHODS: For Temprano, participants were randomly assigned to four groups (deferred ART, deferred ART plus IPT, early ART, or early ART plus IPT). Participants who completed the trial follow-up were invited to participate in a post-trial phase. The primary post-trial phase endpoint was death, as analysed by the intention-to-treat principle. We used Cox proportional models to compare all-cause mortality between the IPT and no IPT strategies from inclusion in Temprano to the end of the follow-up period. FINDINGS: Between March 18, 2008, and Jan 5, 2015, 2056 patients (mean baseline CD4 count 477 cells per μL) were followed up for 9404 patient-years (Temprano 4757; post-trial phase 4647). The median follow-up time was 4·9 years (IQR 3·3-5·8). 86 deaths were recorded (Temprano 47 deaths; post-trial phase 39 deaths), of which 34 were in patients randomly assigned IPT (6-year probability 4·1%, 95% CI 2·9-5·7) and 52 were in those randomly assigned no IPT (6·9%, 5·1-9·2). The hazard ratio of death in patients who had IPT compared with those who did not have IPT was 0·63 (95% CI, 0·41 to 0·97) after adjusting for the ART strategy (early vs deferred), and 0·61 (0·39-0·94) after adjustment for the ART strategy, baseline CD4 cell count, and other key characteristics. There was no evidence for statistical interaction between IPT and ART (pinteraction=0·77) or between IPT and time (pinteraction=0·94) on mortality. INTERPRETATION: In Côte d'Ivoire, where the incidence of tuberculosis was last reported as 159 per 100 000 people, 6 months of IPT has a durable protective effect in reducing mortality in HIV-infected people, even in people with high CD4 cell counts and who have started ART. FUNDING: National Research Agency on AIDS and Viral Hepatitis (ANRS)

Effiziente Lösung reeller Polynomialer Gleichungssysteme

Diese Arbeit beinhaltet {\it geometrische Algorithmen} zur L\"osung reeller polynomialer Gleichungssysteme mit rationalen Koeffizienten, wobei die Polynome eine reduzierte regul\"are Folge bilden (vgl. Abschnitt \ref{abschgeo}). Unter reellem L\"osen verstehen wir hier die Bestimmung eines Punktes in jeder Zusammenhangskomponente einer kompakten glatten reellen Variet\"at $V:=W \cap \R^n$.\\ Im Mittelpunkt steht die Anwendung des f\"ur den algebraisch abgeschlossenen Fall ver\"offentlichten symbolischen geometrischen Algorithmus nach \cite{gh2} und \cite{gh3}. Die Berechenungsmodelle sind {\em Straight--Line Programme} und {arithmetische Netzwerke} mit Parametern in \; \Q. Die Input--Polynome sind durch ein Straight--Line Programm der Gr\"o{\ss}e $L$ gegeben. Eine geometrische L\"osung des Input--Glei\-chungs\-sys\-tems besteht aus einem primitiven Element der Ringerweiterung, welche durch die Nullstellen des Systems beschrieben ist, aus einem mininalen Polynom dieses primitiven Elements, und aus den Parametrisierungen der Koordinaten. Diese Darstellung der L\"osung hat eine lange Geschichte und geht mindestens auf Leopold Kronecker \cite{kron} zur\"uck. Die Komplexit\"at des in dieser Arbeit begr\"undeten Algorithmus erweist sich als linear in $L$ und polynomial bez\"uglich $n, d, \delta$ bzw. \delta \;', wobei $n$ die Anzahl der Variablen und $d$ eine Gradschranke der Polynome im System ist. Die Gr\"o{\ss}en $\delta$ und \delta \; ' sind geometrische Invarianten, die das Maximum der {\em Grade des Inputsystems} und geeigneter {\em polarer Variet\"aten} repr\"asentieren (bzgl. des ({\em geometrischen}) Grades vgl. \cite{he}). Die Anwendung eines Algorithmus \"uber den komplexen Zahlen auf das L\"osen von polynomialen Gleichungen im Reellen wird durch die Einf\"urung polarer Variet\"aten m\"oglich (vgl. \cite{bank}). Die polaren Variet\"aten sind das Kernst\"uck und das vorbereitende Werzeug zur effizienten Nutzung des oben erw\"ahnten geometrischen Algorithmus. Es wird ein inkrementelles Verfahren zur Auffindung reeller Punkte in jeder Zusammenhangskomponente der Nullstellenmenge des Inputsystems abgeleitet, welches einen beschr\"ankten glatten (lokalen) vollst\"andigen Durchschnitt in $\R^n$ beschreibt. Das Inkrement des Algorithmus ist die Kodimension der polaren Variet\"aten. Die Haupts\"atze sind Satz \ref{theorem12} auf Seite \pageref{theorem12} f\"ur den Hyperfl\"achenfall, und Satz \ref{theoresult} auf Seite \pageref{theoresult}, sowie die Aussage in der Einf\"uhrung dieser Arbeit, Seite \pageref{vollres} f\"ur den vollst\"andigen Durchschnitt.This dissertation deals with {\em geometric algorithms} for solving real multivariate polynomial equation systems, that define a reduced regular sequence (cf. subsection \ref{abschgeo}). Real solving means that one has to find at least one real point in each connected component of a real compact and smooth variety $V := W \cap \R^n$. \\ The main point of this thesis is the use of a complex symbolic geometric algorithm, which is designed for an algebraically closed field and was published in the papers \cite{gh2} and \cite{gh3}. The models of computation are {\em straight--line programms} and {\em arithmetic Networks} with parameters in \; \Q. Let the polynomials be given by a division--free straight--line programm of size $L$. A geometric solution for the system of equations given by the regular sequence consists in a {\em primitiv element} of the ring extension associated with the system, a minimal polynomial of this primitive element and a parametrization of the coordinates. This representation has a long history going back to {\em Leopold Kronecker} \cite{kron}. The time--complexity of our algorithms turns out to be linear in $L$ and polynomial with respect to $n, d, \delta$ or $\delta '$, respectively. Here $n$ denotes the number of variables, $d$ is an upper bound of the degrees of the polynomials involved in the system, $\delta$ and $\delta '$ are geometric invariants representing the maximum of the {\em affine (geometric) degree} of the system under consideration and the affine (geometric) degree of suitable {\em polar varieties} (cf. \cite{he} for the ({\em geometric}) degree). The application of an algorithm running in the complex numbers to solve polynomial equations in the real case becomes possible by the introduction of polar varieties (cf. \cite{bank}). The polar varieties introduced for this purpose prove to be the corner--stone and the preliminary tool for the efficient use of the geometric algorithm mentioned above. An incremental algorithm is designed to find at least one real point on each connected component of the zero set defined by the input under the assumption that the given semialgebraic set $V = W \cap \R^n$ is a bounded, smooth (local) complete intersection manifold in $\R^n$. The increment of the new algorithm is the codimension of the polar varieties under consideration. The main theorems are Theorem \ref{theorem12} on page \pageref{theorem12} for the hypersurface case, and Theorem \ref{theoresult} on page \pageref{theoresult} for the complete intersection as well as the statement in the introduction of this thesis on page \pageref{vollres}

Polar Varieties, Real Equation Solving and Data-Structures

In this paper we apply for the first time a new method for multivariate equation solving which was developed in for complex root determination to the real case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input (given in straight--line program representation) and an adequately defined geometric degree of the equation system. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined real (or complex) degree of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above

Equations for Polar Varieties and Efficient Real Elimination

Let V0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1, ..., fp. This paper is devoted to the algorithmic problem of finding efficiently for each connected component of V0 a representative point. For this purpose we exhibit explicit polynomial equations which describe for generic variables the polar varieties of V0 of all dimensions. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1, ..., fp and in a suitably introduced geometric (extrinsic) parameter, called the degree of the real interpretation of the given equation system f1, ..., fp