We define counting #P  classes #P ¡ and in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤-complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥-complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.

Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by f> 0 (or f < 0 or f � = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f −e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set of critical values of the mapping f and the set of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semialgebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach on real-life applications.

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.

We establish a new connection between the two most common traditions in the theory of real computation, the Blum-Shub-Smale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...

Abstract. We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computer-aided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on Sturm-Habicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced object-oriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the public-domain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches. 1

Abstract not found

Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving N P-hardness and #P-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. 1.

In this paper we propose an analytical model of a resilient, tree-based end-node multicast streaming architecture that employs path diversity and forward error correction for improved resilience to node churns and packet losses.

this paper -- we enumerate all 35 di#erent morphologies of QSIC, and characterize each of these morphologies using a signature sequence that can exactly be computed using rational arithmetic. The third problem, not handled here, leads to a lengthy case by case study which depends a lot on the application behind. Consider the intersection curve of two quadrics given by BX = 0, where X = (x, y, z, w) and A, B are 4 4 real symmetric matrices. The characteristic polynomial of (1) and f(#) = 0 is called the characteristic equation of B