Establish the converse implication between first- and second-order SONC-convexity

Determine whether first-order SONC-convexity implies second-order SONC-convexity for general multivariate polynomials of arbitrary even degree.

Background

The paper introduces first-order SONC-convexity through global SONC nonnegativity of the first-order Taylor remainder and second-order SONC-convexity through SONC membership of the Hessian quadratic form. It proves that second-order SONC-convexity does not imply first-order SONC-convexity in general.

For the converse direction, the paper proves only the special case of one variable and degree four, namely $(n,2d)=(1,4)$. Whether the implication extends to the general multivariate and higher-degree setting is explicitly left unresolved.