数值代数几何中的多面体方法
Polyhedral Methods in Numerical Algebraic Geometry
AI总结：
本文提出通过Puiseux级数展开的首项（tropism）作为代数曲线的证书，并将其计算与混合体积相关联，作为更昂贵的witness集计算的预处理阶段。
摘要：
在数值代数几何中，witness集是多项式系统正维解集的数值表示。考虑witness集的渐近性，我们提出了代数曲线的证书。这些证书是曲线在无穷远处开始的Puiseux级数展开的首项。级数中第一项的幂向量是一个tropism。对于真代数曲线，我们将tropism的计算与混合体积的计算联系起来。通过这种关系，tropism和Puiseux级数展开的计算可以用作更昂贵的witness集计算的预处理阶段。具有少量单项式的系统具有更少的孤立解，并且表示其正维解集所需的数据更少。
英文摘要：
In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.
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