On the result of invariance of the closure set of the real projections of the zeros of an important class of exponential polynomials (Q301501)

The paper deals with exponential polynomials \(P(z)=c_1e^{\omega_1 z}+\cdots+c_ne^{\omega_n z},\) \( c_j \in \mathbb{C}\setminus \{0\}, \;\;n\geq 2,\) whose real frequencies \(\omega_j\) are linearly independent over the field of rational numbers. The author considers the set \(R_P:=\overline{\{\Re z: \;\;P(z)=0\}},\) and obtains the following pointwise characterization of \(R_P\). A real number \(\sigma \in R_P\) if and only if NEWLINE\[NEWLINE|c_j|e^{\sigma \omega_ j } \leq \sum_{k=1, k\neq j}^n|c_k|e^{\sigma \omega_k }, \quad j=1,2,\ldots, n.NEWLINE\]NEWLINENEWLINENEWLINEAs a consequence, the author shows the invariance of the set \(R_P\) with respect to the moduli of \(c_j\), \(j=1,2,\ldots, n\). He proves that \(R_P\) is either the closed interval \([a_P,b_P]\), where \(a_P=\inf\{\Re z: P(z)=0\}\) and \(b_P=\sup \{\Re z: P(z)=0\}\), or the union of at most \(n-1\) disjoint nondegenerate closed intervals. In the latter case, the gaps of \(R_P\) are exclusively produced by the equations \(|c_j|e^{\sigma \omega_ j } = \sum_{k=1, k\neq j}^n|c_k|e^{\sigma \omega_k }, \quad j=2,\ldots, n-1,\) having two solutions.NEWLINENEWLINEThe author also investigates the converse of the result on invariance. For polynomials in normalized form \(P(z)=1+m_1e^{\alpha_1 z}+\cdots+m_{n-1}e^{\alpha_{n-1} z}\), if \(R_P=R_Q\), whether it is true or not that the moduli of the coefficients \(P\) and \(Q\) are the same. He obtains an affirmative answer for the cases \(n=2,3,\) and constructs examples showing that it is not true when \(n>3.\)