Certificates for nonnegativity of multivariate integer polynomials under perturbations
Matías R Bender, Khazhgali Kozhasov, Elias Tsigaridas, Chaoping Zhu
TL;DR
The paper introduces an unconditional framework for certifying the global nonnegativity of multivariate integer polynomials by representing them as sums of squares modulo their gradient ideals, without requiring the infimum to be attained or the gradient ideal to be zero-dimensional. It leverages a denominator-free stereographic transformation to obtain coercivity with explicit radius and critical-value bounds, then applies Hanzon–Jibetean perturbations to enforce zero-dimensionality while preserving nonnegativity. Three main algorithms—HJ-SOS-POS, HJ-SOS-NEG, and HJ-SOS-RUR—use these ideas to either certify nonnegativity or produce witness points, with proven single-exponential bit complexity in the number of variables. The work also introduces a new explicit SOS perturbation scheme ensuring that a nonnegative polynomial can be placed inside the SOS cone with concrete perturbation bounds, and provides a thorough complexity analysis and comparison with earlier perturbation methods. Overall, the approach yields principled, computable certificates and witnesses for a broad class of polynomials, expanding the practical applicability of SOS-based nonnegativity certification.
Abstract
We develop a general and unconditional framework for certifying the global nonnegativity of multivariate integer polynomials; based on rewriting them as sum of squares modulo their gradient ideals. We remove the two structural assumptions typically required by other approaches, namely that the polynomial attains its infimum and zero-dimensionality of the gradient ideal. Our approach combines a denominator-free stereographic transformation with a refined variant of the Hanzon--Jibetean perturbation scheme. The stereographic transformation preserves nonnegativity while making the polynomial coercive, with explicit bounds on the radius of positivity and on the nonzero critical values. Subsequently, we apply carefully constructed explicit perturbations that enforce zero-dimensionality of the gradient ideal without altering nonnegativity, allowing us to invoke recent algorithms to derive algebraic certificates or rational witness points. We present three algorithms implementing our framework and analyze their bit complexity in detail, which is single exponential with respect to the number of variables. A second contribution is a new explicit SOS perturbation scheme, which allows us to perturb any nonnegative polynomial in such a way that it can be written as a sum of squares (SOS). In contrast to Lasserre's classical SOS approximation, which guaranties density but currently does not provide an effective control over the perturbation size, we only derive concrete perturbation bounds ensuring that a nonnegative polynomial enters the SOS cone.