It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this paper we provide a counterexample, showing that homogeneous symmetric functions break the pattern. We use a semidefinite program to find a positive semidefinite matrix whose factorization provides an explicit sums of squares decomposition of the polynomial $H_{44} - H_{521}$ as a sum of 41 squares. This rational certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares of other polynomials cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order.

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对称函数不等式的 SOS 反例

众所周知，当定义它们的分区在优势顺序上具有可比性时，对应于各种基的对称函数的差异在非负正交上是非负的。唯一的例外是齐次对称函数的情况，其中只知道分区的优势意味着对称函数的相应差异的非负性。Cuttler、Greene 和 Skandera 在 2011 年推测，逆向也成立，如单项式、初等、幂和和 Schur 基的情况。在本文中，我们提供了一个反例，表明齐次对称函数打破了这种模式。我们使用半定程序来找到一个正半定矩阵，其因式分解提供了多项式 $H_{44} - H_{521}$ 的显式平方和分解为 41 个平方和。这个非负的有理证明推翻了这个猜想，因为作为其他多项式平方和的多项式不能为负，并且因为分区 44 和 521 在优势顺序上是不可比较的。