Let $\mathbb{K}$ be a field of characteristic zero with $\overline{\mathbb{K}}$ its algebraic closure. Given a sequence of polynomials $\mathit{g}=(g_1,\dots ,g_s)\in \mathbb{K}{[x_1,\dots ,x_n]}^s$ and a polynomial matrix $\mathit{F}=\left[f_{i,j}\right]\in \mathbb{K}{[x_1,\dots ,x_n]}^{p\times q}$, with $p\le q$, we are interested in determining the isolated points of $V_p(\mathit{F},\mathit{g})$, the algebraic set of points in $\overline{\mathbb{K}}$ at which all polynomials in $\mathit{g}$ and all $p$-minors of $\mathit{F}$ vanish, under the assumption $n=q-p+s+1$. Such polynomial systems arise in a variety of applications including for example polynomial optimization and real algebraic geometry.

We design a randomized sparse homotopy algorithm for computing the isolated points in $V_p(\mathit{F},\mathit{g})$ which takes advantage of the determinantal structure of the system defining $V_p(\mathit{F},\mathit{g})$. Its complexity is polynomial in the maximum number of isolated solutions to such systems sharing the same sparsity pattern and in some combinatorial quantities attached to the structure of such systems. It is the first algorithm which takes advantage both on the determinantal structure and sparsity of input polynomials.

We also derive complexity bounds for the particular but important case where $\mathit{g}$ and the columns of $\mathit{F}$ satisfy weighted degree constraints. Such systems arise naturally in the computation of critical points of maps restricted to algebraic sets when both are invariant by the action of the symmetric group.

让 $\mathbb{钾}$ 是一个特征为零的域 $\overline{\mathbb{钾}}$它的代数闭包。给定多项式序列$\mathit{G}=(G_1,\dots ,G_{秒})\in \mathbb{钾}{[X_1,\dots ,X_n]}^{秒}$ 和一个多项式矩阵 $\mathit{F}=\left[F_{\mathrm{一世},j}\right]\in \mathbb{钾}{[X_1,\dots ,X_n]}^{磷\times q}$， 和 $磷\le q$，我们有兴趣确定孤立点 ${伏}_{磷}(\mathit{F},\mathit{G})$，点的代数集 $\overline{\mathbb{钾}}$ 其中所有多项式在 $\mathit{G}$ 和所有 $磷$- 未成年人 $\mathit{F}$ 消失，假设下 $n=q-磷+秒+1$. 这种多项式系统出现在各种应用中，包括例如多项式优化和实代数几何。

我们设计了一种随机稀疏同伦算法来计算中的孤立点 ${伏}_{磷}(\mathit{F},\mathit{G})$ 它利用了系统的行列式结构定义 ${伏}_{磷}(\mathit{F},\mathit{G})$. 它的复杂性是共享相同稀疏模式的此类系统的最大孤立解的最大数量的多项式，以及附属于此类系统结构的某些组合量。它是第一个同时利用输入多项式的行列式结构和稀疏性的算法。

我们还推导出特定但重要的情况的复杂性界限，其中 $\mathit{G}$ 和列 $\mathit{F}$满足加权度约束。这种系统在计算受限于代数集的映射的临界点时自然出现，当两者都因对称群的作用而不变时。