2007/07/05 by Christopher J. Hillar, Hillar, Christopher J., Charles R. Johnson +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0707.0712
openalex publication_date 2007/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that the polynomial p(t) = Tr[(A+tB)m] has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to p(t) having positive coefficients for any m and any two n-by-n positive definite matrices. We show that, generally, the question in the real case reduces to that of singular A and B, and this is a key part of our proof.