We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by $τ$. Such a problem is a basic routine in effective real algebraic geometry, used in higher-level algorithms for solving polynomial systems over the reals and finds many applications in sciences. We design a probabilistic algorithm for solving this problem, which is based on reductions to different routines for solving zero-dimensional polynomial systems. It assumes that the input polynomial satisfies sufficiently generic properties (namely, smoothness of its defining hypersurface). This is done through the computations of critical points of well-chosen maps to capture the connected components of the semi-algebraic set under study. We derive a bit complexity estimate for the cost of this algorithm, which is, in terms of the B{é}zout bound d(d -1)^{n-1}, essentially cubic for obtaining parametrisations of the sought-for real points. Moreover, we also consider the case of obtaining rational approximations of those points, which are precise enough to lie in the same connected components as their exact counterparts, which yields a cost that is essentially quartic in the B{é}zout bound. In these complexity estimates, we take into account the degree structure of the input polynomial and its partial derivatives, allowing for a more refined bit complexity when the partial derivative of the input polynomial have degree lower than expected. We also analyse the probability of success of those algorithms. We report on practical experiments, benchmarking with random dense input polynomials as well as polynomials coming from applications, which were out of reach of the state-of-the-art implementations, and hence illustrate the practical efficiency of these new algorithms.

翻译：摘要：我们考虑计算半代数集每个连通分量中的采样点问题，该半代数集由次数为d、有理系数比特规模有界为τ的n元多项式非零或正性定义。此类问题是有效实代数几何中的基本程序，用于实数域上多项式系统求解的高阶算法，并在科学领域有广泛应用。我们设计了一种基于归约至零维多项式系统不同求解程序的概率算法。该算法假设输入多项式满足充分一般性质（即其定义超曲面的光滑性）。通过计算特定映射的临界点来捕捉所研究半代数集的连通分量。我们给出了该算法代价的比特复杂度估计，在Bézout界d(d-1)^{n-1}下，获取所求实点参数化的复杂度本质为立方量级。此外，我们还考虑了获取这些点有理逼近的情形，要求该逼近足够精确以与其精确对应点位于同一连通分量中，此时代价在Bézout界下本质为四次量级。在这些复杂度估计中，我们考虑了输入多项式及其偏导数的次数结构，当输入多项式偏导数次数低于预期时允许更精细的比特复杂度估计。我们还分析了这些算法的成功概率。我们报告了实际实验，包括随机稠密输入多项式和来自应用的多项式基准测试，这些案例超出了现有实现的能力范围，从而展示了新算法的实际效率。