Advances on the Bessis-Moussa-Villani Trace Conjecture
| dc.creator | Hillar, Christopher J. | |
| dc.date | 2005-07-08 | |
| dc.date | 2007-01-06 | |
| dc.date.accessioned | 2026-07-07T07:38:41Z | |
| dc.date.available | 2026-07-07T07:38:41Z | |
| dc.description | A long-standing conjecture asserts that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n \times n$ positive semidefinite Hermitian matrices. The conjecture arises from a question raised by Bessis, Moussa, and Villani (1975) in connection with a problem in theoretical physics. Their conjecture, as shown recently by Lieb and Seiringer, is equivalent to the trace positivity statement above. In this paper, we derive a fundamental set of equations satisfied by $A$ and $B$ that minimize or maximize a coefficient of $p(t)$. Applied to the Bessis-Moussa-Villani (BMV) conjecture, these equations provide several reductions. In particular, we prove that it is enough to show that (1) it is true for infinitely many $m$, (2) a nonzero (matrix) coefficient of $(A+tB)^m$ always has at least one positive eigenvalue, or (3) the result holds for singular positive semidefinite matrices. Moreover, we prove that if the conjecture is false for some $m$, then it is false for all larger $m$. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/math/0507166 | |
| dc.identifier | http://arxiv.org/abs/math/0507166 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121177 | |
| dc.subject | Operator Algebras | |
| dc.subject | Mathematical Physics | |
| dc.subject | 15A24, 15A45, 15A90, 33Cxx, 44A10, 47A50, 47N50, 49J40 | |
| dc.title | Advances on the Bessis-Moussa-Villani Trace Conjecture | |
| dc.type | text |