形状和稳定性假设下格罗布纳基的快速变阶算法
Faster change of order algorithm for Gröbner bases under shape and stability assumptions
摘要：
使用格罗布纳基求解零维多项式系统通常是这样做的，首先，计算一个格罗布纳基的程度反字典顺序，然后计算字典格罗布纳基的顺序变化算法。目前，对于许多通用实例，顺序更改现在占用了整个求解时间的很大一部分。像已知的最快的顺序变化的算法，这项工作的重点是由系统定义的理想满足自然属性，可以恢复在通用坐标的情况下。第一，理想有一个\emph{shape}字典式的Gröbner基。第二，相对于度逆字典序的前导项集合具有\emph{stability}性质;特别地，乘法矩阵可以在输入Gröbner基础上读取。目前最快的算法依赖于这个矩阵的稀疏性。实际上，这种稀疏性是代数结构的结果，可以利用代数结构将矩阵简洁地表示为一元多项式矩阵。我们表明，该矩阵的Hermite规范形式产生所寻求的字典式Gröbner基础，覆盖形状位置的情况下的假设。在对$n \le t$进行适当假设的情况下，算法的复杂度为$O\tilde{~}(t^{ω-1}D)$，其中$n$为变量个数，$t$为矩阵的稀疏性指标，$D$为零维理想的度，$ω$为矩阵乘法的指数.这改进了现有技术的复杂度界限$O\tilde{~}(tD^2)$和$O\tilde{~}(D^ω)$，因为$ω< 3$和$t\le D$。基于msolve库和PML库的实际实验证实了高的实用效益。
英文摘要：
Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a \emph{shape} lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a \emph{stability} property; in particular, the multiplication matrix can be read on the input Gröbner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying $n \le t$, the arithmetic complexity of our algorithm is $O\tilde{~}(t^{ω-1}D)$, where $n$ is the number of variables, $t$ is a sparsity indicator of the aforementioned matrix, $D$ is the degree of the zero-dimensional ideal under consideration, and $ω$ is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds $O\tilde{~}(tD^2)$ and $O\tilde{~}(D^ω)$, since $ω< 3$ and $t\le D$. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.
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