Quantifier elimination over the reals is a central problem in computational real algebraic geometry, polynomial system solving and symbolic computation. Given a semi-algebraic formula (whose atoms are polynomial constraints) with quantifiers on some variables, it consists in computing a logically equivalent formula involving only unquantified variables. When there is no alternate of quantifier, one has a one block quantifier elimination problem. We design a new practically efficient algorithm for solving one block quantifier elimination problems when the input semi-algebraic formula is a system of polynomial equations satisfying some mild assumptions such as transversality. When the input is generic, involves $s$ polynomials of degree bounded by $D$ with $n$ quantified variables and $t$ unquantified ones, we prove that this algorithm outputs semi-algebraic formulas of degree bounded by $\mathcal{D}$ using $O\ {\widetilde{~}}\left (n\ 8^{t}\ \mathcal{D}^{3t+2}\ \binom{t+\mathcal{D}}{t} \right )$ arithmetic operations in the ground field where $\mathcal{D} = n\ D^s(D-1)^{n-s+1}\ \binom{n}{s}$. In practice, it allows us to solve quantifier elimination problems which are out of reach of the state-of-the-art (up to $8$ variables).

中文翻译：

---

正则多项式方程组的更快一格量词消除

在计算实数代数几何，多项式系统求解和符号计算中，消除实数是一个中心问题。给定一个半代数公式（其原子是多项式约束），在某些变量上带有量词，该公式包括计算仅包含未量化变量的逻辑等效公式。当没有量词的替代项时，会有一个整块的量词消除问题。当输入的半代数公式是一个满足多项式假设（例如横向性）的多项式方程组时，我们设计了一种新的实用有效的算法来求解一个块量词消除问题。如果输入是通用的，则涉及度数为$ D $的$ s $多项式，其中包含$ n $个量化变量和$ t $个非量化变量，我们证明了该算法使用$ O \ {\ widetilde {〜}} \ left（n \ 8 ^ {t} \ \ mathcal {D} ^ { 3t + 2} \ \ binom {t + \ mathcal {D}} {t} \ right）$地面字段中的算术运算，其中$ \ mathcal {D} = n \ D ^ s（D-1）^ {n- s + 1} \ \ binom {n} {s} $。在实践中，它使我们能够解决量词消除问题，这是最新技术无法解决的（最多$ 8 $变量）。