Solving parameter-dependent semi-algebraic systems

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.