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INITIAL STEPS IN THE CLASSIFICATION OF MAXIMAL MEDIATED SETS

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 suciently 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.
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Submitted on : Wednesday, August 5, 2020 - 4:34:49 PM
Last modification on : Monday, December 28, 2020 - 10:22:04 AM
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Jacob Hartzer, Olivia Röhrig, Timo de Wolff, Oguzhan Yürük. INITIAL STEPS IN THE CLASSIFICATION OF MAXIMAL MEDIATED SETS. MEGA 2019 - International Conference on Effective Methods in Algebraic Geometry, Jun 2019, Madrid, Spain. ⟨hal-02912343⟩

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