Abstract
The {mathcal {S}}-cone 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 arithmetic-geometric exponentials (SAGE). In this paper, we study the {mathcal {S}}-cone 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 second-order descriptions for rational {mathcal {S}}-cones and their duals.
Highlights
The question to characterize and to decide whether a polynomial or an exponential sum is non-negative occurs in many branches of mathematics and application areas
Generalizing the results of Averkov and of Wang and Magron, we show that rational S-cones and their duals are second-order representable and provide explicit and direct descriptions
Studying the duality theory has been initiated in Chandrasekaran and Shah (2016), Dressler et al (2018b) and Katthän et al (2019)
Summary
The question to characterize and to decide whether a polynomial or an exponential sum is non-negative occurs in many branches of mathematics and application areas. Within the research activities on non-negativity certificates in the last years, the cones of sums of arithmetic-geometric exponentials (SAGE, introduced by Chandrasekaran and Shah 2016) and sums of non-negative circuit polynomials (SONC, introduced by Iliman and de Wolff 2016) have received a lot of attention (see, e.g., Averkov 2019; Dressler et al 2018a; Forsgård and de Wolff 2019; Murray et al 2018, 2019; Wang 2018). These cones build upon earlier work of Reznick (1989). Beyond the specific representability result, the goal of the paper is to offer further insights into the use of the framework of the S-cone as a generalization of SONC and SAGE
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