关于用格罗布纳基计算临界点的复杂性
On the Complexity of Computing Critical Points with Gröbner Bases
摘要：
计算多项式函数$q\in\mathbb Q[X_1,\ldots,X_n]$限制到多项式$f_1,\ldots, f_p\in\mathbb Q[X_1,\ldots, X_n]$的消失轨迹$V\subset\mathbb R^n$的临界点在最优化和实代数几何中的几个应用中是第一重要的。这些点是一个高度结构化的多元多项式方程系统的解决方案，涉及雅可比矩阵的最大未成年人。我们调查的复杂性，解决这个问题，通过使用Gröbner基础算法下的输入多项式的系数的泛型假设。主要结果改进了已知的复杂度界（其取决于最大值$D=\max(deg(f_1),\ldots,deg(f_p),deg(q))$）到取决于度列表$(deg(f_1),\ldots,deg(f_p),deg(q))$的界限：我们证明了Gröbner基的计算可以在$\mathbb Q$的$δ^{O(\log(A)/\log(G))}$算术运算中进行，其中$δ$是临界点上理想消失的代数次数，$A$和$G$是从度序列构造的多集的算术和几何平均。作为一个副产品，我们证明，解决这种通用的优化问题与格罗布纳基需要在最多的$D^{O(n)}$算术运算在$\mathbb Q$，这符合最知名的复杂性约束这个问题。最后，我们用实验说明了这些复杂性结果，证明这些界限与应用程序相关。
英文摘要：
Computing the critical points of a polynomial function $q\in\mathbb Q[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb R^n$ of polynomials $f_1,\ldots, f_p\in\mathbb Q[X_1,\ldots, X_n]$ is of first importance in several applications in optimization and in real algebraic geometry. These points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a Jacobian matrix. We investigate the complexity of solving this problem by using Gröbner basis algorithms under genericity assumptions on the coefficients of the input polynomials. The main results refine known complexity bounds (which depend on the maximum $D=\max(deg(f_1),\ldots,deg(f_p),deg(q))$) to bounds which depend on the list of degrees $(deg(f_1),\ldots,deg(f_p),deg(q))$: we prove that the Gröbner basis computation can be performed in $δ^{O(\log(A)/\log(G))}$ arithmetic operations in $\mathbb Q$, where $δ$ is the algebraic degree of the ideal vanishing on the critical points, and $A$ and $G$ are the arithmetic and geometric average of a multiset constructed from the sequence of degrees. As a by-product, we prove that solving such generic optimization problems with Gröbner bases requires at most $D^{O(n)}$ arithmetic operations in $\mathbb Q$, which meets the best known complexity bound for this problem. Finally, we illustrate these complexity results with experiments, giving evidence that these bounds are relevant for applications.
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