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

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Lukas Katthän, Helen Naumann and Thorsten Theobald
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## 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 $\mathcal {S}$-cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC). We provide a comprehensive characterization of the dual cone of the $\mathcal {S}$-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 $\mathcal {S}$-cone and thus also of its specializations, and we provide a subclass of functions for which non-negativity coincides with membership in the $\mathcal {S}$-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.

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## Additional Information

**Lukas Katthän**- Affiliation: Goethe-Universität, FB 12 – Institut für Mathematik, Postfach 11 19 32, D–60054 Frankfurt am Main, Germany
- Email: katthaen@math.uni-frankfurt.de
**Helen Naumann**- Affiliation: Goethe-Universität, FB 12 – Institut für Mathematik, Postfach 11 19 32, D–60054 Frankfurt am Main, Germany
- Email: naumann@math.uni-frankfurt.de
**Thorsten Theobald**- Affiliation: Goethe-Universität, FB 12 – Institut für Mathematik, Postfach 11 19 32, D–60054 Frankfurt am Main, Germany
- MR Author ID: 618735
- ORCID: 0000-0002-5769-0917
- Email: theobald@math.uni-frankfurt.de
- Received by editor(s): May 3, 2019
- Received by editor(s) in revised form: February 25, 2020, and September 19, 2020
- Published electronically: January 28, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1297-1322 - MSC (2020): Primary 14P05, 90C30; Secondary 52A20, 12D15
- DOI: https://doi.org/10.1090/mcom/3607
- MathSciNet review: 4232225