通过选择来配平：三个简单根处的 NP 完全性
Squaring Up by Selection: NP-Completeness at Three Simple Roots
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中文总结 AI 辅助
本文证明，在超定多项式系统中，通过选择方程进行配平以保持稀疏性的安全性判断是 NP 完全的，即使对于三次系统且只有三个简单根的情况，并揭示了阈值现象与三维匹配问题的联系。
中文摘要 AI 辅助
为了数值求解一个超定多项式系统，通常首先将其配平，通常是用与未知数数量相同的随机线性组合替换给定的方程。这是一个可证明安全的步骤，但它会显著扩大支撑集。另一种选择是保留给定方程中的同样数量。选择保持了稀疏性，但存在几何风险：一个真正的解可能不再是子系统零集的孤立点。我们证明，判断是否存在安全选择是 NP 完全的，即使对于一个次数为三、具有根式理想且恰好有三个简单有理根的显式系统族也是如此。对于强选择，当显式给出非退化有理根时，三次是次数多项式有界时的精确阈值：一个或两个解可归结为拟阵交集，三个解已经导致 NP 完全性。即使没有次数界限，任意长的列表也永远不会使决策问题超出 NP 类。在困难族上，忠实子系统的五个自然概念是重合的，并且每个失败的选择都会明显失败：其零集包含一个通过三个解之一的仿射子空间。一个四次变体表明，代价信息没有帮助：每个可能成功的候选者都有恰好为三的混合体积，而问题仍然是 NP 完全的。该构造将 Karp 的三维匹配问题实现为从给定方程中选择一个方形子系统。
英文摘要
To solve an overdetermined polynomial system numerically, one first makes it square, usually by replacing the given equations with as many random linear combinations as there are unknowns. This is a provably safe step, but it can substantially enlarge the supports. The alternative is to keep that many of the given equations themselves. Selection preserves sparsity but risks geometry: a genuine solution can cease to be an isolated point of the subsystem's zero set. We show that deciding whether a safe choice exists is NP-complete, already for an explicit family of systems of degree three with radical ideal and exactly three simple rational solutions. For strong selection with the nondegenerate rational solutions supplied explicitly, three is the exact threshold when degrees are polynomially bounded: one or two solutions reduce to matroid intersection, three already give NP-completeness. Even without a degree bound, an arbitrarily long list never takes the decision problem beyond NP. On the hard family, five natural notions of a faithful subsystem coincide, and every failing choice fails visibly: its zero set contains an affine subspace through one of the three solutions. A degree-four variant shows that cost information does not help: every candidate that could possibly succeed has mixed volume exactly three, and the problem is NP-complete still. The construction realizes Karp's three-dimensional matching problem as the selection of a square subsystem from the given equations.
发表机构
- CUNY Graduate Center（纽约市立大学研究生中心）
机构由 AI 辅助整理，请以论文原文为准。
