Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators
Abstract
Let ℝ⟨ X,Y⟩ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone Θ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space Z of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on Z they coincide: for f∈ Z, trace-positivity is equivalent to membership in Θ3, to non-negativity of the trace on pairs of 2×2 symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in Θ reduces to a 4×4 positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-(4,2) and (2,4) parts of f; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-(3,3) part is then confined to a rectangle whose sides are the geometric means √(p+q+) and √(p-q-) of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of 2×2 symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of -- a trace-positive polynomial outside Θ -- arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of f is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside Θ.
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BibTeX
@misc{keren2026homogeneous,
title = {{Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators}},
author = {Daniel Keren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12490},
eprint = {2610.12490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12490},
}
RIS
TY - UNPB
TI - Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators
AU - Daniel Keren
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12490
UR - https://arxiv.org/abs/2610.12490
AB - Let ℝ⟨ X,Y⟩ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone Θ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space Z of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on Z they coincide: for f∈ Z, trace-positivity is equivalent to membership in Θ_3, to non-negativity of the trace on pairs of 2×2 symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in Θ reduces to a 4×4 positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-(4,2) and (2,4) parts of f; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-(3,3) part is then confined to a rectangle whose sides are the geometric means √(p_+q_+) and √(p_-q_-) of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of 2×2 symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of -- a trace-positive polynomial outside Θ -- arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of f is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside Θ.
ER -