Collision Positivity for Three-Variable Symmetric Monomial Inequalities
Abstract
Let λ≻γ≻μ be equal-degree exponent partitions with at most three parts, and let Pλ,γ,μ = Jλ+Jμ−2Jγ, where Jν denotes the symmetric monomial orbit sum associated with ν. We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, Pλ,γ,μ(x,y,z)≥0 (x,y,z>0) if and only if Pλ,γ,μ(t,1,1)≥0 (t>0). Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.
Related
- Collision Positivity for Symmetric Orbit-Sum Inequalities in Arbitrary Dimension: A Complete Two-Variable Criterion
- Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants
- When Are Monomial Symmetric Polynomials Hard?
- The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights
- Total positivity and symmetric spaces
BibTeX
@misc{sun2026collision,
title = {{Collision Positivity for Three-Variable Symmetric Monomial Inequalities}},
author = {Jian Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22465},
eprint = {2609.22465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22465},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Collision Positivity for Three-Variable Symmetric Monomial Inequalities
AU - Jian Sun
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22465
UR - https://arxiv.org/abs/2609.22465
AB - Let λ≻γ≻μ be equal-degree exponent partitions with at most three parts, and let P_λ,γ,μ = J_λ+J_μ−2J_γ, where J_ν denotes the symmetric monomial orbit sum associated with ν. We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, P_λ,γ,μ(x,y,z)≥0 (x,y,z>0) if and only if P_λ,γ,μ(t,1,1)≥0 (t>0). Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.
ER -