Topological types of real regular jacobian elliptic surfaces

We present the topological classification of real parts of real regular elliptic surfaces with a real section.Comment: 17 pages, 7 figures, to appear in Geometriae Dedicat

Descartes' Rule of Signs for Polynomial Systems supported on Circuits

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure

Maximally positive polynomial systems supported on circuits

A real polynomial system with support \calW \subset \Z^n is called {\it maximally positive} if all its complex solutions are positive solutions. A support \calW having $n+2$ elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit \calW \subset\Z^n is at most m(\calW)+1, where m(\calW) \leq n is the degeneracy index of \calW. We prove that if a circuit \calW \subset \Z^n supports a maximally positive system with the maximal number m(\calW)+1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of $\Z^n$. In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each $n$ and up to the above action a finite list of circuits \calW \subset \Z^n which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value $1$ or $2$ and make a conjecture in the general case for supports of maximally positive systems

Bounds on the number of real solutions to polynomial equations

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.Comment: 5 page

Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems

Given convex polytopes $P_1,...,P_r$ in $R^n$ and finite subsets $W_I$ of the Minkowsky sums $P_I=\sum_{i \in I} P_i$, we consider the quantity $N(W)=\sum_{I \subset {\bf [}r {\bf ]}} {(-1)}^{r-|I|} \big| W_I \big|$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(W)$ under some assumptions on the family $W=(W_I)$. In particular, we are able to show the nonnegativity of $N(W)$ in some important cases. The quantity $N(W)$ associated with the family defined by $W_I=\sum_{i \in I} W_i$ is called discrete mixed volume of $W_1,...,W_r$. We show that for $r=n$ the discrete mixed volume provides an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $W_1,...,W_n$. We also prove that the discrete mixed volume of $W_1,...,W_r$ is bounded from above by the Kouchnirenko number $\prod_{i=1}^r (|W_i|-1)$. For $r=n$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $W_1,...,W_n$. This conjecture was disproved, but our result shows that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.Comment: 27 pages, small corrections in version

Regions of multistationarity in cascades of Goldbeter–Koshland loops

We consider cascades of enzymatic Goldbeter-Koshland loops (Goldbeter and Koshland in Proc Natl Acad Sci 78(11):6840-6844, 1981) with any number n of layers, for which there exist two layers involving the same phosphatase. Even if the number of variables and the number of conservation laws grow linearly with n, we find explicit regions in reaction rate constant and total conservation constant space for which the associated mass-action kinetics dynamical system is multistationary. Our computations are based on the theoretical results of our companion paper (Bihan, Dickenstein and Giaroli 2018, preprint: arXiv:1807.05157) which are inspired by results in real algebraic geometry by Bihan et al. (SIAM J Appl Algebra Geom, 2018).Fil: Giaroli, Magalí Paola. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Bihan, Frédéric. Université Savoie Mont Blanc. Laboratoire de Mathématiques; FranciaFil: Dickenstein, Alicia Marcela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

UEM : le bout du tunnel ?

C’est maintenant une quasi-certitude : l’UEM entrera en vigueur le 1er janvier 1999. Sa configuration sera définie dès début mai 1998. Depuis le rapport Delors de 1989, sa longue gestation a été rythmée par la crise du SME, la faiblesse de la croissance, la reprise de la hausse du chômage et le gonflement des dettes publiques. Le Traité d’Amsterdam aurait dû définir les nouveaux contours institutionnels de l’Europe (...)