No abstract

a b s t r a c tWe describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f . The algorithm performs O(log(nDκ(f ))) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and κ(f ) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ(f ). The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n, log D and log(κ(f )).

Let I be a compact d-dimensional manifold, let X : I → R be a Gaussian process with regular paths and let FI (u), u ∈ R, be the probability distribution function of sup t∈I X(t).We prove that under certain regularity and nondegeneracy conditions, FI is a C 1 -function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u → +∞. This is a partial extension of previous results by the authors in the case d = 1.Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t) = x, where Z : I → R d is a random field and x is a fixed point in R d . We also give proofs for this kind of formulae, which have their own interest beyond the present application.

We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows. (2000). 65Y20, 65H10. Mathematics Subject Classification

We study the probability distribution F (u) of the maximum of smooth Gaussian fields defined on compact subsets of R d having some geometric regularity.Our main result is a general expression for the density of F . Even though this is an implicit formula, one can deduce from it explicit bounds for the density, hence for the distribution, as well as improved expansions for 1 − F (u) for large values of u.The main tool is the Rice formula for the moments of the number of roots of a random system of equations over the reals.This method enables also to study second order properties of the expected Euler Characteristic approximation using only elementary arguments and to extend these kind of results to some interesting classes of Gaussian fields. We obtain more precise results for the "direct method" to compute the distribution of the maximum, using spectral theory of GOE random matrices.