A Novel Algorithm for Representing Positive Semi-Definite Polynomials as Sums of Squares with Rational Coefficients

2025-10-01

AI Summary

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.

Technology Category

Constraint Satisfaction and Optimization: SatisfiabilityReasoning under Uncertainty: Stochastic OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsResponsible Web: Human-perceived consequences of algorithmic deployment on the webGraph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with floating-point coefficients, our approach ensures that all coefficients in the decomposition remain rational. This is particularly useful in formal verification and symbolic computation, where exact arithmetic is required. We introduce a stepwise reduction technique that transforms a given polynomial into a sum of ladder-like squares while preserving rationality. Experimental results demonstrate the effectiveness of our method compared to existing numerical approaches. This artical is an extension of the following Chinnese paper: HUANG Yong , ZENG Zhenbing , YANG Lu , RAO Yongsheng. An Algorithm to Represent Positive Semi-Definite Polynomials to Sum of Lader-Like Squares of Polynomials with Rational Coefficients (in Chinese). Journal of Systems Science and Mathematical Sciences, 2024, 44(5): 1241-1271 https://doi.org/10.12341/jssms23584CM

Problem

• Develops algorithm for SOS decomposition with rational coefficients
• Ensures exact arithmetic in formal verification applications
• Transforms polynomials into sum of ladder-like squares

Innovation

• Novel algorithm constructs sum-of-squares decomposition
• Ensures all coefficients remain rational numbers
• Uses stepwise reduction for ladder-like squares