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We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $mu(om)$ of (not necessarily compact) basic semi-algebraic sets$omsubsetR^n$. We obtain two monotone (non increasing and non decreasing) sequences of upper and lower bounds $(overline{omega}_d)$, $(underline{omega}_d)$, $dinN$, each converging to $mu(om)$ as $dtoinfty$. For each $d$, computing $overline{omega}_d$ or $underline{omega}_d$reduces to solving a semidefinite program whose size increases with $d$. Some preliminary (small dimension) computational experiments are encouraging and illustrate thepotential of the method. The method also works for any measure whose moments are known and which satisfies Carlemans condition.
Let $Ssubset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this pa
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It
Given any arbitrary semi-algebraic set $X$, any two points in $X$ may be joined by a piecewise $C^2$ path $gamma$ of shortest length. Suppose $mathcal{A}$ is a semi-algebraic stratification of $X$ such that each component of $gamma cap mathcal{A}$ is
We present a theorem of Sard type for semi-algebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, fo