Zero distribution and division results for exponential polynomials
An exponential polynomial of order \(q\) is an entire function of the form \[g(z) = P_1 (z)e^{Q_1 (z)} + \cdots + P_k (z)e^{Q_k (z)},\] where the coefficients \(P_j(z)\), \(Q_j(z)\) are polynomials in \(z\) such that \(\max\{\mathrm{deg}(Q_j)\} = q\). \par The authors present many motivational examples and discuss the zero distribution of exponential polynomials. \par It is known that the majority of the zeros of a given exponential polynomial lies in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. The condition has the following form. \par Theorem 3.1. Let \(g\) be an exponential polynomial of the form \[g(z)=H_1(z)e^{w_1z^q}+\cdots+H_m(z)e^{w_mz^q},\] where the coefficients \(H_j(z)\) are exponential polynomials of growth \(\rho(H_j) \le q-p\) for some integer \(1\le p \le q\). Suppose that the points \(w_1,\dots,w_m\) are either corner points or interior points of the convex hull co(W). (In other words, all points \(w_1,\dots,w_m\) that are on the boundary of co(W) are corner points.) Then the majority of the zeros of \(g\) are simple in the sense that \[N_{\ge 2}(r,1/g)=O(r^{q-p}+\log r),\] where \(N_{\ge 2}(r,1/g)\) is the integrated counting function of the zero of \(g\) in \(|z|\le r\) with multiplicity at least \(2\). \par Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by \textit{J. F. Ritt} [Trans. Am. Math. Soc. 31, 680--686 (1929; JFM 55.0212.01)] in the case \(q = 1\) with constant coefficients. Ritt's result is closely related to Shapiro's conjecture that has remained open since 1958. To formulate the division result the authors define two function classes for \(p\in\mathbb{N}\cup\{0\}:\) \[P_p=\{f: f \text{ is an exponential polynomial and }\rho(f)\le p\},\] \[R_p=\{f/g: f,g\in P_p, \ g\not\equiv 0\}.\] Theorem 4.1. Let \(n\), \(m\), \(q\in \mathbb{N}\), and \(p\le q-1\) be a non-negative integer. Set \[g(z)=\sum_{j=1}^m a_j(z)e^{\mu_jz^q} \text{ and } h(z) =\sum_{j=1}^n b_j(z)e^{\nu_j z^q},\] where \(a_j\), \(b_j\in P_p\) are non-vanishing. If \(f(z) = g(z)/h(z)\) is entire, then \(f\) too is an exponential polynomial, but with coefficients in \(R_p\).
scientific article; zbMATH DE number 6976583