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

Zhenbing Zeng, Yong Huang, Lu Yang, Yongsheng Rao

TL;DR

This work tackles representing positive semidefinite polynomials with rational coefficients as sums of squares, avoiding floating-point artifacts for exact arithmetic. It introduces the L-SOS algorithm, which constructs a degree-descending SOS p(x)=q_d(x)^2+...+q_0^2 via a structured lower-triangular factor L partitioned into border, core, and diagonal blocks, solved through a triangular coefficient system to guarantee rational solutions. The authors extend the approach to univariate and multivariate cases, including a project-and-lift framework for multivariate SOS, and demonstrate numerous examples with exact rational decompositions while discussing limitations in more complex scenarios. The method offers a practical, exact alternative to numerical SOS techniques, with implications for formal verification and symbolic computation and a roadmap for scaling to higher-degree problems.

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

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

TL;DR

This work tackles representing positive semidefinite polynomials with rational coefficients as sums of squares, avoiding floating-point artifacts for exact arithmetic. It introduces the L-SOS algorithm, which constructs a degree-descending SOS p(x)=q_d(x)^2+...+q_0^2 via a structured lower-triangular factor L partitioned into border, core, and diagonal blocks, solved through a triangular coefficient system to guarantee rational solutions. The authors extend the approach to univariate and multivariate cases, including a project-and-lift framework for multivariate SOS, and demonstrate numerous examples with exact rational decompositions while discussing limitations in more complex scenarios. The method offers a practical, exact alternative to numerical SOS techniques, with implications for formal verification and symbolic computation and a roadmap for scaling to higher-degree problems.

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

Paper Structure

This paper contains 7 sections, 3 theorems, 243 equations.

Table of Contents

  1. Motivation and Background
  2. SOS representation of degree-strictly-descending polynomials
  3. An algorithm for constructing DDP-SOS representation
  4. SOS of Rational Coefficients and the Cholesky Factorization
  5. Examples of DDP SOS of rational coefficients
  6. Rational SOS for multi-variate polynomials
  7. Conclusion

Key Result

theorem thmcountertheorem

Any semi-positive definite polynomial $p(x)$ od degree $2d$ can be expressed as the sum of squares of as follows: where $q_k(x)\, (1 \leq k \leq d)$ is a zero polynomial or a polynomial of degree $k$, and $q_0(x) = q_0$ is a real number.

Theorems & Definitions (6)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof