Initial steps in the classification of maximal mediated sets. (March 2022)

• Record Type: Journal Article
• Authors: Hartzer, Jacob; Röhrig, Olivia; de Wolff, Timo; Yürük, Oğuzhan
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 404
• Page End: 425
• Publication Date: 2022-03
• Subjects: 14P10 -- 52B20 -- 11E25; Lattice -- Maximal mediated set -- Nonnegativity -- Sum of nonnegative circuit polynomials -- Sums of squares; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.013
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Maximal mediated sets (MMS), introduced by Reznick, are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares. In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices are isomorphic if and only if the simplices generate the same lattice up to permutations. Furthermore, we generalize a result of Iliman and the third author. Practically, we fully characterize the MMS for all simplices of sufficiently small dimensions and maximal 1-norms. In particular, we experimentally prove a conjecture by Reznick for 2 dimensional simplices up to maximal 1-norm 150 and provide indications on the distribution of the density of MMS.