construct sos certificates

This paper addresses degree automation of the Sum-of-Squares (SoS) proof system: given a fixed degree (d), can SoS refutations for systems of multivariate polynomials—particularly those outside the scope of Raghavendra–Weitz’s framework—be constructed automatically in polynomial time? It specifically targets unsatisfiable constraint satisfaction problem (CSP) instances, marking the first extension of SoS degree automation to refutation. Method: Leveraging tools from algebraic proof complexity, Nullstellensatz-based analysis, and algebraic CSP modeling, the authors integrate bit-complexity considerations with algebraic geometry techniques to derive new criteria for SoS degree automation. Contribution/Results: They establish novel sufficient conditions for SoS degree automation; rigorously separate classes of CSP instances that admit vs. resist SoS refutation; and design the first polynomial-time SoS refutation algorithms for multiple concrete constraint families—surpassing prior frameworks limited to satisfiable instances with large solution spaces.

This paper addresses the problem of certifying positivity of univariate polynomials with rational coefficients over real intervals. We propose an efficient algebraic certification framework based on weighted sums of squares (SOS) and perturbed SOS representations. Our method explicitly constructs a two-square decomposition for nonnegative univariate polynomials—a first such result—and reveals structural connections between their roots, Karlin points, and T-systems. We design a hybrid algorithm integrating numerical approximation, symbolic computation, and bit-complexity analysis to generate structured SOS certificates explicitly. The algorithm achieves bit complexity Õ_B(d³ + d²τ) and certificate size Õ(d²τ), improving upon prior work by a factor of O(d). An open-source Maple implementation demonstrates substantial gains in numerical accuracy, computational efficiency, and certificate verifiability.

This work investigates whether symmetry inherently causes exponential growth in the bit-width of coefficients within Sum-of-Squares (SoS) proofs, thereby undermining computational feasibility. Focusing on Archimedean symmetric polynomial optimization problems, we integrate the SoS hierarchy, semidefinite programming relaxations, and group representation-theoretic symmetry analysis. We establish the first separation of two orthogonal sources of SoS complexity: proof degree and coefficient bit-size. Our results show that while symmetry may necessitate higher-degree SoS proofs, it does *not* intrinsically induce large-bit coefficients; low-degree symmetric SoS proofs admit compact numerical representations. This refutes the common assumption that symmetry inevitably leads to coefficient blow-up. The findings provide a new theoretical foundation for automated SoS solving and lower-bound analysis, advancing our understanding of computational tractability in symmetric algebraic proof systems. (149 words)

This paper addresses the rational sum-of-squares (SOS) decomposition problem for semidefinite polynomials with rational coefficients. Conventional approaches rely on floating-point arithmetic, inherently compromising coefficient rationality and introducing rounding errors. To overcome this, we propose a constructive algorithm grounded in a *staircase square structure* and *rational algebraic reduction*: it hierarchically constructs rational square terms while enforcing exact rational arithmetic at every step, thereby ensuring globally exact, rounding-free SOS decompositions. Our method is the first to systematically support rational-coefficient SOS representations in arithmetic-critical domains such as formal verification and symbolic computation. Experimental evaluation on multiple benchmark instances demonstrates stable generation of exact rational solutions, significantly enhancing both the reliability and applicability of symbolic SOS computation.

This work studies local certification of $P_5$-free graphs—graphs containing no induced path on five vertices—with the goal of designing a subquadratic distributed verification scheme. To overcome the $Omega(n^2)$ certificate-size bottleneck of conventional approaches, we propose a novel certification framework based on structural graph decomposition, modular encoding, and localized label propagation. Our method achieves, for the first time, an $O(n^{3/2})$ upper bound on the certificate size. Each vertex verifies $P_5$-freeness using only its local neighborhood and an $O(n^{3/2})$-bit global certificate, with verification completing in constant rounds. We provide a rigorous theoretical analysis proving that this $O(n^{3/2})$ bound is asymptotically tight, thereby establishing an optimal complexity threshold for local certification of sparse graph properties. This result significantly advances the state of the art in distributed graph property testing.

This study addresses the computational tractability and exact rational solving challenges of Sum-of-Squares optimization over the Boolean hypercube. To overcome these limitations, this work constructs enhanced moment semidefinite programs by exploiting truncated vanishing identities, and introduces geometric realization and transfer theorems that integrate bit-complexity frameworks, Gröbner bases, the ellipsoid method, and spectral coefficient bounds. The main contributions demonstrate that polynomial systems closed under Min/Max operations are solvable in polynomial time with constructible certificates. Furthermore, it establishes a polynomial-time algorithm for searching rational proofs at fixed relaxation degrees, achieving exact rational feasibility and arbitrary-precision optimization. These results effectively break through the longstanding semidefinite programming bottleneck for Boolean Sum-of-Squares hierarchies.

本文解决了证书复杂度与近似度之间关系的问题，通过构造特定布尔函数族证明了证书复杂度可以比近似度大四次方，改进了之前的结果。

研究确定了多维交叉多面体的多项式逼近的度数-失真权衡，通过非负形式和平方和形式分析，解决了在不同维度下的逼近问题。

研究了通过Hajós连接和顶点识别方法解决图的k-着色问题的Nullstellensatz证书最小系数度，提出了一种构建特定类型4-临界图的方法。

该研究解决了多项式演算中的理想成员问题，通过基于代数方法的约简框架，对特定约束语言下的问题可解性进行了探讨。