The reformulation of the Bessis–Moussa–Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient
α
m
,
k
(
A
,
B
)
of
t
k
in the polynomial
Tr
(
A
+
tB
)
m
, with
A
,
B
...positive semidefinite matrices, is nonnegative for all
m
,
k
. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of
m
,
k
the method is successful, obtaining a complete determination when either
m
or
k
is odd.
A long-standing conjecture asserts that the polynomial
p
(
t
)
=
Tr
(
A
+
tB
)
m
has nonnegative coefficients whenever m is a positive integer and A and B are any two
n
×
n positive semidefinite ...Hermitian matrices. The conjecture arises from a question raised by Bessis et al. D. Bessis, P. Moussa, M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys. 16 (1975) 2318–2325 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. Finally, we outline a general program to settle the BMV conjecture that has had some recent success.
We show that all the coefficients of the polynomial
are nonnegative whenever
m
≤13 is a nonnegative integer and
A
and
B
are positive semidefinite matrices of the same size. This has previously been ...known only for
m
≤7. The validity of the statement for arbitrary
m
has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.
Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani conjecture that all coefficients of the polynomial p(t) = tr((A +tr B)
m
) are non-negative whenever A and B are any two ...positive semidefinite matrices of the same size. We will show that for all m∈ℕ the coefficient of t
4
in p(t) is non-negative, using a connection to sums of Hermitian squares of non-commutative polynomials which has been established by Klep and Schweighofer. This implies by a well-known result of Hillar that the coefficients of t
k
are non-negative for 0 ≤ k ≤ 4, and by symmetry as well for m ≥ k ≥ m − 4.
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