Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.

Finding one point on each semi-algebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Although numerous studies have been done on the subject, only a small number of efficient implementations exist.In this paper, we propose a new efficient and practical algorithm for computing such points. By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zerodimensional systems whose zeros contain at least one point on each semi-algebraically connected component of the studied variety, without any assumption either on the variety (smoothness or compactness for example) or on the system of equations which define it.From the output of our algorithm, one can then apply, for each computed zerodimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions. We report some experiments using a set of pure exact methods. The practical efficiency of our method is due to the fact that we do not apply any infinitesimal deformations, unlike the existing methods based on a similar strategy.

Use of food resources by herbivores depends on intrinsic constraints, essentially body size and morpho-physiological characteristics, which determine the range of foods they tolerate and environmental constraints, such as seasonality and interspecific interactions, which determine the availability of resources. We analysed a collection of rumen contents samples from sympatric populations of red deer Cervus elaphus and roe deer Capreolus capreolus and tested several theoretical predictions relating to the impact of intrinsic and environmental constraints on diet composition, diversity and similarity. Red deer consumed more slowly digestible, fibrous forage than roe deer and had a more diverse diet throughout the year, which supports predictions deriving from specific body size and morpho-physiological characteristics. In conformity with the optimal foraging theory, both species consumed more slowly digestible forage in times of low food availability (i.e. during winter) than during the rest of the year. An increase in diet similarity in winter, along with predictions from the theory on competitive interaction processes, led us to assume that food resources were not limiting and that exploitative competition between red and roe deer was unlikely in our study area. We underline the importance of studies of the use of food resources by sympatric herbivores in answering applied ecological questions at the local scale, and we suggest that the Euclidean geometrical approach we used is particularly well suited for the analysis of resource matrices, a common end-product of long-term field data gathering on the feeding habits of animals.

Since the recent spread of highly pathogenic (HP) H5N1 subtypes, avian influenza virus (AIV) dispersal has become an increasing focus of research. As for any other bird-borne pathogen, dispersal of these viruses is related to local and migratory movements of their hosts. In this study, we investigated potential AIV spread by Common Teal (Anas crecca) from the Camargue area, in the South of France, across Europe. Based on bird-ring recoveries, local duck population sizes and prevalence of infection with these viruses, we built an individual-based spatially explicit model describing bird movements, both locally (between wintering areas) and at the flyway scale. We investigated the effects of viral excretion duration and inactivation rate in water by simulating AIV spread with varying values for these two parameters. The results indicate that an efficient AIV dispersal in space is possible only for excretion durations longer than 7 days. Virus inactivation rate in the environment appears as a key parameter in the model because it allows local persistence of AIV over several months, the interval between two migratory periods. Virus persistence in water thus represents an important component of contamination risk as ducks migrate along their flyway. Based on the present modelling exercise, we also argue that HP H5N1 AIV is unlikely to be efficiently spread by Common Teal dispersal only.

International audienceFour methods for solving polynomial systems by means of triangular sets are presented and implemented in a unified way. These methods are those of Wu (1987), Lazard (1991), Kalkbrener (1991) and Wang (1993b). They are compared on various examples with the emphasis on efficiency, conciseness and legibility of the output