消失理想与布尔域上平方和的计算可解性

中文摘要 AI 辅助

基于Raghavendra-Weitz的位复杂度框架和Gribling-Polak-Slot的矩-SOS准则，我们研究了布尔多项式系统中截断消失恒等式的有效利用。一个完整且可构造的恒等式空间给出了一个增广矩SDP，该SDP可以通过精确有理可行性和任意加性精度进行优化。我们给出一个自包含的几何实现：一个显式仿射约简和一个布尔求值单纯形提供了有理椭球所需的半径界。恒等式空间可以直接构造，或从给定的分次Groebner基（包括在更高截断度下提供的基）中提取。一个显式传递定理将这一增广公式与原始系统联系起来。双边的SoS推导从证书中消除了附加的等式公理，具有受控的度和系数增长，并意味着投影的更高层矩松弛包含在增广体中。结合谱系数界，这给出了具有加性扰动的有理证明的多项式时间搜索。对于最小闭线性系统，传播在固定度下以多项式时间构造恒等式空间，并为每个度-t基元素的正负号给出度-(4t+4)证书。因此，对于固定d，只要f >= 0具有度-2d证明，就可以在多项式时间内找到f + epsilon >= 0的有理度-(8d+4)证明。增广的度-2d矩SDP可以在多项式时间内以精确有理可行性进行优化，并与原始的度-(8d+4)松弛进行比较。布尔互补对最大闭系统给出相同结果，包括广义打包和覆盖。因此，最小闭系统为一般准则提供了具体应用。所有复杂度界均在图灵模型中。

英文摘要

Building on the bit-complexity framework of Raghavendra-Weitz and the moment-SOS criteria of Gribling-Polak-Slot, we study the effective use of truncated vanishing identities over Boolean polynomial systems. A complete, constructible identity space gives an augmented moment SDP that can be optimized with exact rational feasibility and arbitrary additive accuracy. We give a self-contained geometric implementation: an explicit affine reduction and a simplex of Boolean evaluations supply the radius bounds required by rational ellipsoids. The identity space can be constructed directly or extracted from a supplied graded Groebner basis, including one supplied at a higher truncation degree. An explicit transfer theorem connects this augmented formulation to the original system. Two-sided SoS derivations eliminate the added equality axioms from certificates, with controlled degree and coefficient growth, and imply containment of a projected higher-level moment relaxation in the augmented body. Together with spectral coefficient bounds, this gives polynomial-time search for rational proofs with an additive perturbation. For Min-closed linear systems, propagation constructs the identity space in polynomial time at fixed degree and gives degree-(4t+4) certificates for both signs of each degree-t basis element. Consequently, a rational degree-(8d+4) proof of f + epsilon >= 0 can be found in polynomial time for fixed d whenever f >= 0 has a degree-2d proof. The augmented degree-2d moment SDP can be optimized in polynomial time with exact rational feasibility and a comparison to the original degree-(8d+4) relaxation. Boolean complementation gives the same results for Max-closed systems, including generalized packing and covering. Min-closed systems thus provide a concrete application of the general criteria. All complexity bounds are in the Turing model.

发表机构

• SUPSI-IDSIA（瑞士应用科学与艺术大学 - IDSIA）

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