Zeros of a one-parameter family of harmonic trinomials

Abstract

It is well known that complex harmonic polynomials of degree n may have more than n zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.

On the number of roots for harmonic trinomials

“…In [2], the authors provide a way to count the roots for the family of harmonic trinomials when a = 1, c = −1 and b ∈ (0, ∞). Moreover, the maximum number of roots for this family is n+2m, see Theorem 1.1 in [2].…”

“…In [2], the authors provide a way to count the roots for the family of harmonic trinomials when a = 1, c = −1 and b ∈ (0, ∞). Moreover, the maximum number of roots for this family is n+2m, see Theorem 1.1 in [2]. As a consequence of our main result Theorem 1.2 we obtain Corollary 1.4 which yields that any harmonic trinomial has at most n + 2m roots.…”

Zeros of a Two-parameter Family of Harmonic Quadrinomials

“…Recently, Brilleslyper et al [3] studied on the number of zeros of harmonic trinomials of the form p c (z) = z n + cz k − 1 where 1 ≤ k ≤ n − 1, n ≥ 3, c ∈ R + , and gcd(n, k) = 1. They showed that the number of zeros of p c (z) changes as c varies and proved that the distinct number of zeros of p c (z) ranges from n to n + 2k.…”

Location of the Zeros of Certain Complex-Valued Harmonic Polynomials

“…For an overview of the topic, see Duren [12] and Dorff and Rolf [13]. It was shown by Bshouty et al [14] that there exists a complex-valued harmonic polynomial f = h + g, such that h is an analytic polynomial of degree n, g is an analytic polynomial of degree m < n and f has exactly n 2 zeros counting with multiplicities in the field of complex numbers, C. We are motivated by the work of Brilleslyper et al [15], Kennedy [7], and Dehmer [2] on the number and location of zeros of trinomials. Recently, Brilleslyper et al [15] studied the number of zeros of harmonic trinomials of the form p c (z…”

“…Among other things, they used the argument principle for harmonic function that can be formulated as a direct generalization of the classical result for analytic functions (Duren et al [16]). An interesting open problem raised in [15] was finding where the zeros of this trinomials are located and deciding whether the value of c affects the zero inclusion regions of p c (z).…”