Zeros of a Family of Complex-Valued Harmonic Trinomials

On the number of roots for harmonic trinomials

“…Recently, the corresponding geometric sectors as in (II) for harmonic trinomials of the form (1.2) with A = 1, B ∈ C \ {0}, C = −1 has been derived in [22]. For references about location, counting, geometry, and lower/uppers bounds for the moduli of roots for harmonic polynomials including probabilistic approaches and numerical experiments, we refer to [1,11,12,13,22,23,24,25,26,27,29,30,31,32,36,37,38,39,40,54,55,56] and the references therein.…”

Zeros of Convex Combinations of Elementary Families of Harmonic Functions

“…More specifically, the sign of the order of a zero or a pole depends on whether it is in the sense-preserving region (positive order) or the sense-reversing region (negative order). See [11] for a more detailed discussion of why this convention makes sense. For this paper, we take the following to be our definitions of the order of a zero and the order of a pole.…”

“…They have sought to determine the maximum and minimum number of zeros, how the number of zeros varies with the coefficients, and the location of these zeros. See [7][8][9][10][11][12][13][14][15]. We discuss several of their results in some detail because they are the motivation for the current work.…”

“…In [9,10], the authors considered a one-parameter family of trinomials. Although they determined precisely how the number of zeros varies with the parameter, a corollary of their main theorem is as follows:…”

“…Brooks and Lee [12] carried out a similar detailed analysis of an analogous oneparameter family of harmonic functions with poles. A corollary of their main theorem is as follows:…”