Let $\mathbb{K}$ be a field of characteristic zero and let $\bar{\mathbb{K}}$ be an algebraic closure of $\mathbb{K}$. Consider a sequence of polynomials $\mathit{G}=(g_1,\dots ,g_s)$ in $\mathbb{K}[X_1,\dots ,X_n]$ with $s<n$, a polynomial matrix $\mathit{F}=[f_{i,j}]\in \mathbb{K}{[X_1,\dots ,X_n]}^{p\times q}$, with $p\le q$ and $n=q-p+s+1$, and the algebraic set $V_p(\mathit{F},\mathit{G})$ of points in $\bar{\mathbb{K}}$ at which all polynomials in G and all p-minors of F vanish. Such polynomial systems appear naturally in polynomial optimization or computational geometry.

We provide bounds on the number of isolated points in $V_p(\mathit{F},\mathit{G})$ depending on the maxima of the degrees in rows (resp. columns) of F and we design probabilistic homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V_p(\mathit{F},\mathit{G})$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

让 $\mathbb{ķ}$ 成为特征零的字段，让 $\stackrel{〜}{\mathbb{ķ}}$ 是...的代数闭包 $\mathbb{ķ}$。考虑多项式序列$\mathit{G}=（G_{\mathrm{1个}}，\dots ，G_s）$ 在 $\mathbb{ķ}[X_{\mathrm{1个}}，\dots ，X_{ñ}]$ 与 $s<ñ$，一个多项式矩阵 $\mathit{F}=[F_{\mathrm{一世}，Ĵ}]\in \mathbb{ķ}{[X_{\mathrm{1个}}，\dots ，X_{ñ}]}^{p\times q}$，带有 $p\le q$ 和 $ñ=q-p+s+\mathrm{1个}$和代数集 $V_p（\mathit{F}，\mathit{G}）$ 的点数 $\stackrel{〜}{\mathbb{ķ}}$G的所有多项式和F的所有p-次项都消失了。这样的多项式系统自然出现在多项式优化或计算几何中。

我们提供了对中孤立点数的限制 $V_p（\mathit{F}，\mathit{G}）$根据F的行（或列）中度的最大值，我们设计了概率同伦算法来计算这些点。这些算法利用了系统定义的确定性结构$V_p（\mathit{F}，\mathit{G}）$。特别地，算法在隔离点数量的界限上是多项式的时间上运行。