Here we provide code which can explicitly verify the sum of squares counterexample appearing in the article https://arxiv.org/abs/1909.00081.
License: MIT
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An SOS counterexample to an inequality of symmetric functions
This document verifies the fact that a certain symmetric polynomial can be written as a sum of squares. This provides a counterexample to a certain conjecture relating nonnegativity and the dominance (or majorization) partial order on partitions. See https://arxiv.org/abs/1909.00081 for more information. Our symmetry-adapted SDP returned a numerical matrix (with floating point entries). Since we had reduced the problem size using representation theory of the symmetric group, and also knowledge of the real zeros of our polynomial, we were able to successfully round the floating point entries into (exact) rational numbers.
We load this matrix , whose entries are rational numbers, as well as a permutation matrix which reorders the columns and rows so that we can apply naive decomposition below. In order to apply decomposition, it is sometimes required to reorder the rows and columns in order to avoid zero pivots.
Below we apply decomposition to extract pairs where is the pivot entry and is a vector. These pairs can be used to write .
Below we create a vector of monomials with total degree in three variables. The permutation matrix loaded above reorders this basis appropriately to match the reordering of the matrix .
Below we sum the squares. We use each vector to create a polynomial , which we square and sum with positive coefficient . The resulting polynomial displayed below is therefore a sum of squares with positive coefficients.
Below we create the symmetric polynomial , evaluated at to check nonnegativity on the nonnegative orthant, as explained in https://arxiv.org/abs/1909.00081. The nonnegativity of this polynomial provides the counterexample to the conjecture.
Below we verify that our previous sum of squares reproduces the desired polynomial.