On the lower bound of the refined Bessis--Moussa--Villani conjecture: proof of the cases S_{6,3} and S_{7,3} for 3x3 matrices
Abstract
For Hermitian matrices α,β let An,m(eα,eβ) be the average of the traces of all words with n letters eα and m letters eβ. The refined Bessis--Moussa--Villani conjecture contains the lower bound An,m(eα,eβ)≥Tr enα+mβ (Tr denotes the trace), which for n=m=1 is the Golden--Thompson inequality. We prove this lower bound for all n+m≤7 when the matrices have size at most three, and for min(n,m)≤2 in every dimension. In all these cases we prove the stronger inequality An,m(A,B)≥Tr(An/mB)m for positive semidefinite A,B, or its mirror image with the roles of A and B exchanged; the lower bound of the refined conjecture follows from the Araki--Lieb--Thirring inequality. The cases with three letters B in dimension three, (n,m)=(3,3) and (4,3), are reduced by an exact identity and the inequality of arithmetic and geometric means to a polynomial inequality in two variables, which is certified in exact rational arithmetic by Bernstein expansion on three boxes. The certificate is a short program in the ancillary files.
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BibTeX
@misc{hagele2026lower,
title = {{On the lower bound of the refined Bessis--Moussa--Villani conjecture: proof of the cases S_6,3 and S_7,3 for 3x3 matrices}},
author = {Daniel Hägele},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00129},
eprint = {2611.00129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00129},
}
RIS
TY - UNPB
TI - On the lower bound of the refined Bessis--Moussa--Villani conjecture: proof of the cases S_{6,3} and S_{7,3} for 3x3 matrices
AU - Daniel Hägele
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00129
UR - https://arxiv.org/abs/2611.00129
AB - For Hermitian matrices α,β let A_n,m(e^α,e^β) be the average of the traces of all words with n letters e^α and m letters e^β. The refined Bessis--Moussa--Villani conjecture contains the lower bound A_n,m(e^α,e^β)≥Tr e^nα+mβ (Tr denotes the trace), which for n=m=1 is the Golden--Thompson inequality. We prove this lower bound for all n+m≤7 when the matrices have size at most three, and for min(n,m)≤2 in every dimension. In all these cases we prove the stronger inequality A_n,m(A,B)≥Tr(A^n/mB)^m for positive semidefinite A,B, or its mirror image with the roles of A and B exchanged; the lower bound of the refined conjecture follows from the Araki--Lieb--Thirring inequality. The cases with three letters B in dimension three, (n,m)=(3,3) and (4,3), are reduced by an exact identity and the inequality of arithmetic and geometric means to a polynomial inequality in two variables, which is certified in exact rational arithmetic by Bernstein expansion on three boxes. The certificate is a short program in the ancillary files.
ER -