We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x = (x_1, \ldots, x_n)$, $V\subset \mathbb{C}^{t+n}$ be the algebraic set defined by $f$ and $\pi$ be the projection $(y, x) \to y$. Under the assumptions that $f$ admits finitely many complex roots for generic values of $y$ and that the ideal generated by $f$ is radical, we solve the following problem. On input $f$, we compute semi-algebraic formulas defining semi-algebraic subsets $S_1, \ldots, S_l$ of the $y$-space such that $\cup_{i=1}^l S_i$ is dense in $\mathbb{R}^t$ and the number of real points in $V\cap \pi^{-1}(\eta)$ is invariant when $\eta$ varies over each $S_i$. This algorithm exploits properties of some well chosen monomial bases in the algebra $\mathbb{Q}(y)[x]/I$ where $I$ is the ideal generated by $f$ in $\mathbb{Q}(y)[x]$ and the specialization property of the so-called Hermite matrices. This allows us to obtain compact representations of the sets $S_i$ by means of semi-algebraic formulas encoding the signature of a symmetric matrix. When $f$ satisfies extra genericity assumptions, we derive complexity bounds on the number of arithmetic operations in $\mathbb{Q}$ and the degree of the output polynomials. Let $d$ be the maximal degree of the $f_i$'s and $D = n(d-1)d^n$, we prove that, on a generic $f=(f_1,\ldots,f_n)$, one can compute those semi-algebraic formulas with $O^~( \binom{t+D}{t}2^{3t}n^{2t+1} d^{3nt+2(n+t)+1})$ operations in $\mathbb{Q}$ and that the polynomials involved have degree bounded by $D$. We report on practical experiments which illustrate the efficiency of our algorithm on generic systems and systems from applications. It allows us to solve problems which are out of reach of the state-of-the-art.

中文翻译：

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通过Hermite矩阵求解多项式方程组的参数系统

我们设计了一种新的算法来求解参数系统，该系统对于参数的泛型值具有有限的许多复杂解。更准确地说，让$ f =（f_1，\ ldots，f_m）\ subset \ mathbb {Q} [y] [x] $与$ y =（y_1，\ ldots，y_t）$和$ x =（x_1，\ ldots，x_n）$，$ V \ subset \ mathbb {C} ^ {t + n} $是由$ f $定义的代数集，而$ \ pi $是到$$的投影$（y，x）。假设$ f $接受$ y $的泛型值的有限复数根，并且$ f $产生的理想是激进的，我们可以解决以下问题。在输入$ f $上，我们计算定义$ y $空间的半代数子集$ S_1，\ ldots，S_l $的半代数公式，以使$ \ cup_ {i = 1} ^ l S_i $密集于$当$ \ eta $在每个$ S_i $上变化时，\ mathbb {R} ^ t $和$ V \ cap \ pi ^ {-1}（\ eta）$中的实际点数不变。该算法利用了代数$ \ mathbb {Q}（y）[x] / I $中一些精选单项式基的性质，其中$ I $是$ f $在$ \ mathbb {Q}（y）中生成的理想值[x] $和所谓的Hermite矩阵的专业化性质。这使我们能够通过编码对称矩阵签名的半代数公式来获得集合$ S_i $的紧凑表示。当$ f $满足额外的泛型假设时，我们得出$ \ mathbb {Q} $中算术运算的数量和输出多项式的阶数的复杂性界限。假设$ d $是$ f_i $的最大程度，并且$ D = n（d-1）d ^ n $，我们证明，在通用$ f =（f_1，\ ldots，f_n）$上，可以用$ O ^〜（\ binom {t + D} {t} 2 ^ {3t} n ^ {2t + 1} d ^ {3nt + 2（n + t）+1}计算那些半代数公式）$在$ \ mathbb {Q} $中的运算，并且所涉及的多项式的度数受$ D $限制。我们报告了一些实际实验，这些实验说明了我们的算法在通用系统和应用程序系统上的效率。它使我们能够解决最新技术无法解决的问题。