Page 107 - Textos de Matemática Vol. 39

NONNEGATIVE INVERSE EIGENVALUE PROBLEM 97
conjectured that if f(x) has integer coefficients and the power sums sk are positive and in addition satisfy
μ(d)sk/d ≥ 0 1≤d|k
for all positive integers k, then σ with sufficiently many zeros appended can be realized by a nonnegative integer matrix. This conjecture was proved in a very impressive paper by Kim, Ormes and Roush [10].
Using the techniques of classical analysis, especially delicate examination of the behaviour of power series, they show that a nonnegative polynomial matrix B(t) can be found satisfying
c(t) = det(I − B(t)).
They show that there exists such a B(t) of a very specific form:
⎡qr(t) 1 0 ... 0 ⎤
⎢ar−2 qr−1(t) 1 ... . ⎥ B(t) = ⎢ar−3 0 ... ... 0 ⎥,
⎢⎣. .......1⎥⎦ a0(t) 0 ... 0 q1(t)
where the entries qj(t), al(t) are integer polynomials with nonnegative coeffi- cients and zero constant terms. The corresponding form of c(t) is:
r
c(t)= (1−qj(t))−a0(t)−a1(t)q1(t)−a2(t)q1(t)q2(t)−...−ar(t)q1(t)...qr−2(t).
j=1
This leads to a realization of σ with sufficient zeros appended by an integer matrix of a block lower Hessenberg form.
The question of whether every realizable spectrum σ can be realized by a nonnegative lower Hessenberg matrix appears to be open. The work in [10] leads one to consider realizability by nonnegative matrices of the form:
⎡C1 10...0⎤ ⎢Ar−2 C2 1 ... . ⎥
(3.3) F = ⎢Ar−3 0 ... ... 0 ⎥, ⎢⎣. .......1⎥⎦
A0(t) 0 ... 0 Cr
where the matrices Cj are nonnegative companion matrices.
Since each Cj has exactly one positive real root, one should choose r to be at least the number of positive entries in σ. We do not know any realizable