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Mathematics of Computation

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


Authors: Lukas Katthän, Helen Naumann and Thorsten Theobald
Journal: Math. Comp. 90 (2021), 1297-1322
MSC (2020): Primary 14P05, 90C30; Secondary 52A20, 12D15
DOI: https://doi.org/10.1090/mcom/3607
Published electronically: January 28, 2021
MathSciNet review: 4232225
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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
Article copyright: © Copyright 2021 American Mathematical Society