Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gau{\ss} map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: - The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. - A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. -The fact that this inequality is sharp in the non-singular case.

Cet article est une introduction a la geometrie algebrique et a certaines de ses applications. Des rappels d'algebre commutative (groupes, anneaux, ideaux, corps des reels et des complexes) constituent une introduction. Est abordee d'abord l'etude des racines de polynomes a une variable, resultant, discriminant, suites de Sturm et algorithmes d'isolation des racines reelles. Geometrie algebrique dans le plan affine, dans le plan projectif (cas reel et complexe), point singuliers surfaces de Riemann et theoreme de Harnack sont ensuite traites. Pour finir, le principe de quelques algorithmes d'elimination (intersection de deux courbes planes reelles, passage d'une representation parametrique a une equation intrinseque) est donne.

Soit Δ⊂𝐑 2 un polygone convexe à sommets entiers ; G. Mikhalkin a défini les « courbes de Harnack » (définies par un polynôme de support contenu dans Δ et plongées dans la surface torique correspondante) et montré leur existence (via la « méthode du patchwork de Viro ») ainsi que l’unicité de leur type topologique plongé (qui est determiné par Δ). Le but de cet article est de montrer un résultat analogue pour la lissification (smoothing) d’un germe de branche réelle plane (C,O) analytique réelle. On définit pour cela une classe de smoothings dite « Multi-Harnack » à l’aide de la résolution des singularités constituée d’une suite de g éclatements toriques, si g est le nombre de paires de Puiseux de la branche (C,O). Un smoothing multi-Harnack est réalisé de la manière suivante : à chaque étape de la résolution (en commençant par la dernière) et de manière successive, un smoothing « De Harnack » (au sens de Mikhalkin) intermédiaire est obtenu par la méthode de Viro. On montre alors l'unicité du type topologique de tels smoothings. De plus, on peut supposer ces smoothings « multi-semi-quasi homogènes » ; on montre alors que des propriétés métriques (« multi-taille » des ovales) de tels smoothings sont caractérisées en fonction de la classe d’équisingularité de (C,O) et que réciproquement ces tailles caractérisent la classe d’équisingularité de la branche.

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler's appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.

The local Harnack inequality bounds from above the number of ovals which can appear in a small perturbation of a singular point. As is known, there are singular points for which this bound is not sharp. We show that Harnack inequality is sharp in any complex topologically equisingular class: every real plane curve singular point is complex deformation equivalent to a real singularity for which Harnack inequality is sharp. For semi-quasi-homogeneous and some other singularities we exhibit a real deformation with the same property. A refined Harnack inequality and its sharpness are discussed also.

Let Z be a germ of a singular real analytic vector field at O∈R2. We give conditions on the multiplicity and the Milnor number of Z which imply that the foliation defined by Z has a characteristic orbit or an analytic invariant curve with the hypothesis that Z is a real generalized curve. Then it is proved that for a non-dicritical real generalized curve, the multiplicity mod2 is invariant under bilipshitz homeomorphisms preserving foliations.

the aij and bij being real or complex. This vector field is a deformation of the vector field x∂y −y∂x whose trajectories are concentric circles around 0. We prove in this paper a precise version of the following assertion: for any compact K in the space of the (aij ,bij ), there exist a number p(q) and a neighborhood U(q,K) of 0 such that for (a,b) ∈ K , either 0 is again a center of Wa,b (i.e., 0 is an elliptic nondegenerate singular point of W , and W is integrable near 0) or Wa,b has at most p(q) limit cycles in U(q,K). The local sixteenth Hilbert problem consists of finding explicit expressions for U(q,K) and p(q). This problem is solved only for q = 2 by the socalled Bautin theorem (see [B], [Ya]). Bautin considered the Poincaré first return map around the origin restricted to a line with coordinate X as a series Fz(X) in X with coefficients depending on the parameters z = (aij ,bij ). The limit cycles correspond to the zeroes of Fz(X)−X. Given a series

Let ξ be a polynomial vector field on C n with coefficients of degree d and P be a polynomial of degree p.We are interested in bounding the multiplicity of a zero of a restriction of P to a non-singular trajectory of ξ, when P does not vanish identically on this trajectory.Bounds doubly exponential in terms of n are already known ([9, 5, 10]).In this paper, we prove that, when n = 3, there is a bound of the form p + 2p(p + d -1) 2 .In Control Theory, such a bound can be used to give an estimate of the degree of nonholonomy for a system of polynomial vector fields (this degree expresses the level of Lie-bracketing needed to generate the tangent space at each point).

ABSTRACTSince 1935, France has experienced an evolution ‐ from legislation aimed at groundwater protection to regulations concerning water‐management incentives for increased use, in selected parts of acquifers, without causing over‐abstraction. During the 1960s, the protection of users became the main priority, and the water agencies have now become powerful bodies in the combat against water pollution and towards aquifer rehabilitation. The 1992 Water Act provided a balanced management of water resources (a) taking into account the water resource, surface‐water and groundwater, and (b) establishing a Water Directorate within the Ministry of the Environment. Reference is made to major achievements in the field of groundwater management and protection in terms of quantity and quality.

Let = (V 1 ; : : : ; V s ) be a system made with vector elds V 1 ; : : : ; V s in R n whose coordinates are polynomials of degree d. To such a system is associated the control system _ x = P u i V i . It is proven that the degree of nonholonomy of such a system is bounded by a function (n; d) (2nd) n2 (n+1) . In the case n = 2, we have (2; d) 6d 2 2d+ 2.

The maximum of the degree of nonholonomy of the control system attached to a kinematic model for the car with n trailers is Fn+3 where Fn is the nth term of the Fibonacci’s sequence (F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn, n ≥ 0). This value is obtained if and only if each trailer (except for the last one) is perpendicular to the previous one.

In this paper we describe an algorithm of construction of a roadmap for a compact semi-algebraic setS ⊂ Rn, which is similar to the algorithm of [3], but which is simpler in the sense that it does not need the use of Whitney stratifications, and more general because it accepts as input any compact semi-algebraic set. The complexity of this algorithm is measured in terms of the numberk of polynomials definingS, their maximum degreed, and the number of variablesn. With respect to those measures, our algorithm runs in time\(\left( {kd} \right)^{O\left( {n^3 } \right)} \). The goal of this paper is not a strengthening of the previous results (similar bounds of complexity are obtained in [5] and [13], even for a non-compact set), but more a new approach, and we hope the efficiency of the algorithm.

1. B-splines 2. Spline curves and Bezier curves 3. Interpolation and complements 4. Spline surfaces 5. Triangulations 6. Notions of real algebraic geometry Plates Bibliography Index.