Positive Univariate Polynomials: SOS certificates, algorithms, bit complexity, and T-systems

2025-10-02

AI Summary

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.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge 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 deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of) $mathbb{R}$, given as weighted sums of squares (SOS) of rational polynomials. We build on the algorithm of Chevillard, Harrison, Joldes, and Lauter~cite{chml-usos-alg-11}, which we call usos. For a polynomial of degree~$d$ and coefficient bitsize~$τ$, we show that a rational weighted SOS representation can be computed in $widetilde{mathcal{O}}_B(d^3 + d^2 τ)$ bit operations, and the certificate has bitsize $widetilde{mathcal{O}}(d^2 τ)$. This improves the best-known bounds by a factor~$d$ and completes previous analyses. We also extend the method to positivity over arbitrary rational intervals, again saving a factor~$d$. For univariate rational polynomials we further introduce emph{perturbed SOS certificates}. These consist of a sum of two rational squares approximating the input polynomial so that nonnegativity of the approximation implies that of the original. Their computation has the same bit complexity and certificate size as in the weighted SOS case. We also investigate structural properties of these SOS decompositions. Using the classical fact that any nonnegative univariate real polynomial is a sum of two real squares, we prove that the summands form an interlacing pair. Their real roots correspond to the Karlin points of the original polynomial, linking our construction to the T-systems of Karlin~cite{Karlin-repr-pos-63}. This enables explicit computation of such decompositions, whereas only existential results were previously known. We obtain analogous results for positivity over $(0,infty)$ and thus over arbitrary real intervals. Finally, we present an open-source Maple implementation of usos and report experiments on diverse inputs that demonstrate its efficiency.

Problem

• Computing certificates of positivity for univariate rational polynomials
• Developing efficient algorithms for weighted SOS representations
• Analyzing structural properties of SOS decompositions and T-systems

Innovation

• Computes rational weighted SOS certificates efficiently
• Introduces perturbed SOS certificates for approximations
• Links SOS decompositions to T-systems via interlacing