Certificates for nonnegativity of multivariate integer polynomials under perturbations

TL;DR

This paper introduces a new framework for certifying the global nonnegativity of multivariate integer polynomials by combining a stereographic transformation with explicit perturbations, removing previous structural assumptions and providing effective SOS certificates.

Contribution

It develops an unconditional, general approach for nonnegativity certification of multivariate polynomials, including new algorithms and explicit perturbation schemes that improve upon existing SOS methods.

Findings

Framework removes structural assumptions like infimum attainment and zero-dimensionality.

Algorithms have single exponential bit complexity in the number of variables.

New SOS perturbation scheme guarantees a polynomial can be expressed as a sum of squares.

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.