Nonnegativity of differences of term-normalised complete homogeneous symmetric functions: degrees 8, 9 and 10 in three variables, decided exactly

Abstract

For a partition λ of d let h_λ be the product of complete homogeneous symmetric polynomials in n variables and H_(n,λ)=h_λ/h_λ(1ⁿ) its term-normalised form. Cuttler, Greene and Skandera proved that dominance of partitions implies H_(n,λ) ≤ H_(n,μ) on the nonnegative orthant and conjectured the converse; Heaton and Shankar refuted the converse at n=3, d=8 with a sum of 41 squares, reported that their search found "many other counterexamples, in degrees 8, 9, and 10", and stated that those were left in floating point. We settle, exactly and in full, the three degrees they name, for n=3. Of the 270 ordered questions "is H_(3,μ)-H_(3,λ) ≥ 0 on orth(3)?" posed by the 135 dominance-incomparable pairs of degree 8, 9 or 10, exactly 58 have the answer yes and 212 have the answer no; none is left undecided. Each positive answer is certified by between 42 and 63 nonnegative integers — the coefficients of the difference after the substitution x₁=a, x₂=a+b, x₃=a+b+c, that is, one step of the classical successive difference substitution method — and each negative answer by one of only five rational points. We also correct one printed coefficient in the expansion displayed by Heaton and Shankar, prove that their counterexample survives in four variables, and observe that the number of counterexamples is not monotone in n. Nonnegativity, not sum-of-squares representability, is what is decided here. All statements are machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   1b54fa82763606dad3456e3dca16746698f6ba8dfdcc6cbf50b3f1e82fd8a896