Initial Steps in the Classification of Maximal Mediated Sets

TL;DR

This paper advances the understanding of maximal mediated sets (MMS) in lattice polytopes, providing theoretical classifications and practical characterizations for small-dimensional simplices, with implications for polynomial sums of squares.

Contribution

It offers a classification of MMS for simplices, proves isomorphism conditions, and characterizes MMS for small simplices, extending previous results and testing conjectures.

Findings

MMS of simplices are isomorphic iff they generate the same lattice.

Complete characterization of MMS for small-dimensional simplices.

Experimental validation of Reznick's conjecture for 2D simplices up to norm 150.

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.