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 Scone, 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 Scone and thus also of its specializations, and we provide a subclass of functions for which non-negativity coincides with membership in the S-cone.Moreover, 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.
The Scone provides a common framework for cones of polynomials or exponential sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmeticgeometric exponentials (SAGE). In this paper, we study the Scone and its dual from the viewpoint of second-order representability. Extending results of Averkov and of Wang and Magron on the primal SONC cone, we provide explicit generalized secondorder descriptions for rational Scones and their duals.
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