We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in [2] to arbitrary semialgebraic sets.All previous algorithms proposed for this problem have doubly exponential complexity.

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Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the padic worlds. We show how the p-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free p-adic polynomial. Moreover, we show that this algorithm runs in O(d 2 log 3 d)-time for a random p-adic polynomial of degree d. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into p-adic numerical algebraic geometry.

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We present a version of Smale's α-theory for ultrametric fields, such as the p -adics and their extensions, which gives us a multivariate version of Hensel's lemma.

We present a version of Smale's α-theory for ultrametric fields, such as the p-adics and their extensions, which gives us a multivariate version of Hensel's lemma. Hensel's lemma [4, §3.4] gives us sufficient condition for lifting roots mod p k to roots in Z p . Alternatevely, Hensel's lemma gives us sufficient conditions for Newton's method convergence towards an approximate root. Unfortunately, in the multivariate setting, versions of Hensel's lemma are scarce [2]. However, in the real/complex world, Smale's α-theory [3] gives us a clean sufficient criterion for deciding if Newton's method will converge quadratically. In the p-adic setting, Breiding [1] proved a version of the γ-theorem, but he didn't provide a full α-theory. In this short communication, we provide an ultrametric version of Smale's α-theory for square systems-initially presented as an appendix in [5]-, together with an easy proof.In what follows, and for simplicity 1 , F is a non-archimedian complete field of characteristic zero with (ultrametric) absolute value | | and P n,d [n] the set of polynomial maps f : F n → F n where f i is of degree d i . In this setting, we will consider on F n the ultranorm given by x := max{x 1 , . . . , x n }, its associated distance dist(x, y) := x − y , and on k-multilinear maps A : (F n ) k → F q the induced ultranorm, which is given by

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