INITIAL STEPS IN THE CLASSIFICATION OF MAXIMAL MEDIATED SETS
Résumé
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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