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：我们考虑在由n元d次多项式（有理系数比特大小为$τ$）的非零或正性定义的半代数集的每个连通分量中计算样本点的问题。该问题是实代数几何中的基本例程，用于实数域上多项式系统求解的高层算法，并在科学领域有广泛应用。我们设计了一个概率算法解决该问题，其核心是通过约化到求解零维多项式系统的不同例程来实现。该算法假设输入多项式满足足够通用的性质（即其定义超曲面的光滑性）。我们通过计算精心选取的映射的临界点来捕获所研究半代数集的连通分量。我们推导了该算法代价的比特复杂度估计，就Bézout界$d(d-1)^{n-1}$而言，获取所寻求实点的参数化本质上需要三次复杂度。此外，我们还考虑了获取这些点的有理逼近的情况，这些逼近需足够精确以与精确点位于相同连通分量中，这导致代价本质上为Bézout界的四次方。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，当输入多项式的偏导数次数低于预期时，可得到更精细的比特复杂度。我们还分析了这些算法的成功概率。我们报告了实际实验，对随机稠密输入多项式以及来自应用的多项式进行了基准测试，这些案例超出了现有实现的能力范围，从而验证了新算法的实际效率。