用Hermite矩阵求解实数上的参数多项式方程组
Solving parametric systems of polynomial equations over the reals through Hermite matrices
摘要：
我们设计了一个新的算法，用于求解参数系统的一般值具有许多复杂的解决方案。更准确地说，设$f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$具有$y = (y_1, \ldots, y_t)$和$x = (x_1, \ldots, x_n)$，$V\subset \mathbb{C}^{t+n}$是由$f$定义的代数集，$π$是投影$(y, x) \to y$。在假设$f$对于$y$的一般值有2/3个复根，并且$f$生成的理想是根的条件下，我们解决了下面的问题。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，使得当$η$在每个$S_i$上变化时，$\cup_{i=1}^l S_i$在$\mathbb{R}^t$中是稠密的，并且$V\cap π^{-1}(η)$中的实点数是不变的。该算法利用了代数$\mathbb{Q}(y)[x]/I$中一些精心选择的单项式基的性质，其中$I$是由$\mathbb{Q}(y)[x]$中的$f$生成的理想，以及所谓的Hermite矩阵的特殊化性质。这使我们能够获得紧凑的表示集$S_i$的半代数公式编码的对称矩阵的签名。当$f$满足额外的通用性假设，我们推导出的$\mathbb{Q}$的算术运算的数量和输出多项式的程度的复杂性界限。设$d$是$f_i$和$D = n(d-1)d^n$的最大次数，证明了在一般的$f=(f_1,\ldots,f_n)$上，可以用$\mathbb{Q}$中的$O^~( \binom{t+D}{t}2^{3t}n^{2t+1} d^{3nt+2(n+t)+1})$运算计算那些半代数公式，并且所涉及的多项式的次数以$D$为界.我们报告的实际实验，说明了我们的算法的效率，通用系统和系统的应用。它使我们能够解决超出最先进水平的问题。
英文摘要：
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 $π$ 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 $V\cap π^{-1}(η)$ is invariant when $η$ 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.
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