We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are considered as parameters and we assume that when specialising them generically, the set of common complex solutions, to the obtained equations, is finite. We consider the problem of real root classification for such parameter-dependent problems, i.e. identifying the possible number of real solutions depending on the values of the parameters and computing a description of the regions of the space of parameters over which the number of real roots remains invariant. We design an algorithm for solving this problem. The formulas it outputs enjoy a determinantal structure. Under genericity assumptions, we show that its arithmetic complexity is polynomial in both the maximum degree $d$ and the number $s$ of the input inequalities and exponential in $nt+t^2$. The output formulas consist of polynomials of degree bounded by $(2s+n)d^{n+1}$. This is the first algorithm with such a singly exponential complexity. We report on practical experiments showing that a first implementation of this algorithm can tackle examples which were previously out of reach.

翻译：我们考虑$\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$中的多项式方程与不等式系统，其中$\boldsymbol{x} = (x_1, \ldots, x_n)$，$\boldsymbol{y} = (y_1, \ldots, y_t)$。将$\boldsymbol{y}$未知元视为参数，并假设当对其作一般性特化时，所得方程组的公共复解集合是有限的。针对此类参数相关问题的实根分类——即根据参数取值确定实解的可能个数，并计算参数空间中实根数目保持不变的区域的描述——我们设计了一个求解算法。该算法输出的公式具有行列式结构。在一般性假设下，我们证明其算术复杂度关于输入不等式的最大次数$d$和个数$s$呈多项式增长，关于$nt+t^2$呈指数增长。输出公式由次数不超过$(2s+n)d^{n+1}$的多项式构成。这是首个具有单指数复杂度的算法。我们报告了实际实验的结果，表明该算法的首个实现能够处理此前无法解决的实例。