Cylindrical algebraic decomposition (CAD) plays an important role in the field of real algebraic geometry and many other areas. As is well-known, the choice of variable ordering while computing CAD has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called "the perfect elimination orderings"), naturally preserve chordality, which is an important property on sparsity of variables. Experimentation suggests that if the associated graph of the polynomial system in question is chordal (\emph{resp.}, is nearly chordal), then a perfect elimination ordering of the associated graph (\emph{resp.}, of a minimal chordal completion of the associated graph) can be a good variable ordering for the CAD computation. That is, by using the perfect elimination orderings, the CAD computation may produce a much smaller full set of projection polynomials than by using other naive variable orderings. More importantly, for the complexity analysis of the CAD computation via a perfect elimination ordering, a so-called $(m,d)$-property of the full set of projection polynomials obtained via such an ordering is given, through which the "size" of this set is characterized. This property indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller "size". This is well consistent with the experimental results, hence the perfect elimination orderings with lower elimination tree height are further recommended to be used in the CAD projection.

中文翻译：

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利用弦结构选择圆柱代数分解的变量序

圆柱代数分解（CAD）在实际代数几何和许多其他领域中起着重要作用。众所周知，在计算CAD时选择变量顺序对计算的时间和内存使用以及计算的采样点数量有很大的影响。在本文中，我们指出，典型的CAD算法（如果针对特殊类型的变量排序（称为“完美消除命令”）执行）自然保留了弦性，这是变量稀疏性的重要属性。实验表明，如果所讨论的多项式系统的关联图是弦的（\ emph {resp。}，几乎是弦的），则关联图的完美消除顺序（\ emph {resp。}，相关图的最小和弦补全）可以是CAD计算的良好变量排序。也就是说，与使用其他朴素的变量排序相比，通过使用理想的消除顺序，CAD计算可以生成更小的全套投影多项式。更重要的是，对于通过完美消除顺序进行的CAD计算的复杂性分析，给出了通过这种顺序获得的投影多项式的全部集合的所谓的$（m，d）$-属性，其中“这套的特点是。此属性表明，当相应的完全消除树的高度较低时，全套投影多项式也趋向于具有较小的“大小”。这与实验结果非常吻合，