2022/06/26 by Chaudhury, Srijonee Shabnam
#15A18 15B36 15B57 11C08 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2206.12871
Let S be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and T to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such that all non-zero entries are square-roots of some positive integers and the matrices satisfy a certain cycle condition. In this paper, for any n × n matrix A ∈ S ∪ T and any k ∈ ℕ we find a general lower bound for Tr2k(A), i.e, the sum of 2k-th power of eigenvalues of A, which depends on n as well as some other variables. In particular, we obtain the best possible lower bound for Tr2(A) that is 6n - 5. As a strong outcome of this result we show that the smallest limit point of Tr2(A) = (Tr2(A))/(n) is 6. This is a solution of an analogue of ``Schur - Siegel - Smyth trace problem" for characteristic polynomials of matrices in S ∪ T. We also obtain a lower bound of smallest limit point of Tr2k(A) for any positive integer k > 1 and for the same set of matrices. Furthermore, we exhibit that the famous results of Smyth on density of absolute trace measure and absolute trace-2 measure of totally positive integers are also true for the set of symmetric integer connected positive definite matrices.