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.

Reznick 引入的最大介导集 (MMS) 是具有偶数顶点的积分多胞体中格点的显着子集。AGI 形式的牛顿多面体和非负电路多项式的 MMS 确定这些多项式是否为平方和。

在本文中，我们从理论上和实践上对 MMS 进行了初步分类。从理论上讲，我们证明了单形的 MMS 是同构的，当且仅当单形产生相同的格直至排列。此外，我们概括了 Iliman 和第三作者的结果。实际上，我们完全表征了所有足够小维度和最大 1 范数的单纯形的 MMS。特别是，我们通过实验证明了 Reznick 对最大 1-范数 150 的二维单纯形的猜想，并提供了 MMS 密度分布的指示。