A unified framework of SAGE and SONC polynomials and its duality theory

Abstract

We introduce and study a cone which consists of a class of generalized polynomial functions and which provides a common framework for recent non-negativity certificates of polynomials in sparse settings. Specifically, this $S$-cone generalizes and unifies recent cones of polynomials that establish non-negativity upon the arithmetic-geometric inequality (SAGE cone, SONC cone). We provide a comprehensive characterization of the dual cone of the $S$-cone, which even for its specializations provides novel and projection-free descriptions. As applications of this result, we give an exact characterization of the extreme rays of the $S$-cone and thus also of its specializations, and we provide a subclass of functions for which non-negativity coincides with membership in the $S$-cone. \nMoreover, we derive from the duality theory an approximation result of non-negative univariate polynomials and show that a SONC analogue of Putinar s Positivstellensatz does not exist even in the univariate case.