Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem come from monomial substitutions into the arithmetic-geometric inequality. In 1989, the second author gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms (Math. Ann., (283), 431--464), involving the arrangement of lattice points in the simplex whose vertices were the $n$-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.

中文翻译：

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关于中介单纯形的注释

在平方和和希尔伯特第 17 个问题的研究中出现的许多齐次多项式来自单项式代换到算术几何不等式。1989 年，第二作者给出了这样一种形式的充要条件，它具有形式的平方和的表示（Math. Ann., (283), 431--464），涉及格点在顶点是替换中使用的指数的 $n$-元组的单纯形。此外，有人声称但未证明任何此类单纯形的足够大的膨胀也将满足此条件。这个简短说明的目的是在希尔伯特第 17 个问题的研究和格点单纯形的研究中证明这一主张，并为结果提供进一步的背景。