三次二分Holant问题带复代数权重的二分法
A Dichotomy for Cubic Bipartite Holant Problems with Complex Algebraic Weights
- Google（谷歌）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文对带复代数权重的三次二分Holant问题给出精确复杂度二分法，识别出可处理签名族（秩一、广义相等及仿射变换）并证明其余为#P-hard，含实数交集刻画。
AI中文摘要：
我们对每个固定的复代数对称布尔三元签名$f$，对$\operatorname{Holant}(f\mid=_3)$的精确求值进行分类。输入是一个三次二分多重图：一侧的每个顶点携带$f$，另一侧的每个顶点携带三元相等，且没有可自由使用的辅助签名。可处理的签名恰好是秩一张量、广义相等、以及六个仿射签名的保相等立方根对角变换，连同非零缩放和反转。每个其他签名在多项式时间图灵归约下给出$\\#\mathrm{P}$-难问题。我们还确定了精确的实数交集：它由与有理分类相同的可处理族组成，参数为实代数。证明在每条预言机实例的两侧都精确保持度为三。通过插值提取的秩一矩阵仅在未使用因子被三元组吸收后提供一个一元签名。在复数上，这种吸收有三个例外射影方向。我们将此约束与射影矩阵群轨道、显式三元替换以及对有限射影阶的穷尽处理相结合。阶为三和五的情况包括精确多项式证书；证书恒等式和有理算术验证器作为补充材料提供。
英文摘要：
We classify the exact evaluation of $\operatorname{Holant}(f\mid=_3)$ for every fixed complex algebraic symmetric Boolean ternary signature $f$. An input is a cubic bipartite multigraph: every vertex on one side carries $f$, every vertex on the other side carries ternary equality, and no auxiliary signatures are freely available. The tractable signatures are precisely rank-one tensors, generalized equalities, and equality-preserving cube-root diagonal transformations of six affine signatures, together with nonzero scalings and reversal. Every other signature gives a $\#\mathrm{P}$-hard problem under polynomial-time Turing reductions. We also identify the exact real intersection: it consists of the same tractable families as in the rational classification, with real algebraic parameters. The proof preserves degree exactly three on both sides of every oracle instance. A rank-one matrix extracted by interpolation supplies one unary signature only after its unused factor has been absorbed in triples. Over the complex numbers this absorption has three exceptional projective directions. We combine this constraint with projective matrix-group orbits, explicit ternary replacements, and an exhaustive treatment of finite projective orders. The cases of orders three and five include exact polynomial certificates; the certificate identities and a rational-arithmetic verifier are supplied as supplementary material.
