{'fraction_non_alphanumeric': 0.053368044256426946, 'fraction_numerical': 0.015131793036121055, 'mean_word_length': 3.773980582524272, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Location of the Zeros of Certain Complex-Valued Harmonic Polynomials

Finding an approximate region containing all the zeros of analytic polynomials is a well-studied problem. But the numb er of the zeros and regions containing all the zeros of complex-valued harmonic polynomials is relatively a fresh research area. It is well known that all the zeros of analytic trinomials are enclosed in some annular sectors that take into account the magnitude of the coefficients. Following Kennedy and Dehmer, we provide the zero inclusion regions of all the zeros of complex-valued harmonic polynomials in general, and in particular, we bound all the zeros of some families of harmonic trinomials in a certain annular region.


Introduction
The location of the zeros of analytic polynomials has been studied by many researchers and we refer the reader to Brilleslyper and Schaubroeck [1], Dhemer [2], Frank [3], Gilewicz and Leopold [4], Howell and Kyle [5], Johnson and Tucker [6], Kennedy [7], and Melman [8]. Recently, researching the number of the zeros of general analytic trinomials and the regions in which the zeros are located has become of interest due to their application in other fields. For example, the roots of the trinomial equations can be interpreted as the equilibrium points of unit masses that are located at the vertices of two regular concentric polygons centered at the origin in the complex plane (Szabo [9]). A useful result in determining the location of the zeros of analytic polynomials is the argument principle. Recall that if f is analytic inside and on positively oriented rectifiable Jordan curve C and f (z) = 0 on C, then the winding number of the image curve f (C) about the origin , 1 2π ∆ C argf (z), equals the total number of zeros of f in D, counted according to multiplicity where D is a plane domain bounded by C. In 1940 Kennedy [7] showed that the roots of analytic trinoimial equations of the form z n + az k + b = 0, where ab = 0, have certain bounds to their respective absolute values. In 2006, Dehmer [2] proved that all the zeros of complex-valued analytic polynomials lie in certain closed disk. In 2012, Melman [8] investigated the regions in which the zeros of analytic trinomials of the form p(z) = z n − αz k − 1, with integers n ≥ 3 and 1 ≤ k ≤ n − 1 with gcd(k, n) = 1, and α ∈ C, lie. He determined the zero inclusion regions for the following cases: a) for any value of |α|; b) |α| > σ(n, k); and c) |α| < σ(n, k), where σ(n, k) = n n−k ( n−k k ) k n . In each cases, Melman provided useful information on the location of the zeros of p.
One area of investigation that has recently become of interest is the number and location of zeros of complex-valued harmonic polynomials. In 1984 Clunie and Sheil-Small [10] introduced the family of complex-valued harmonic functions f = u + iv defined in the unit disk D = {z : |z| < 1}, where u and v are real harmonic in D. A continuous function f = u + iv defined in a domain G ⊂ C is harmonic in G if u and v are real harmonic in G. In any simply connected subdomain of G, we can decompose f as f = h + g, where g and h are analytic and g denotes the function z → g(z)(see Duren [11]). This family of complex-valued harmonic functions is a generalization of analytic mappings studied in geometric function theory, and much research has been done investigating the properties of these harmonic functions. 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) = 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. 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).
In this paper, we first look at general harmonic polynomials and locate the zeros by using some results obtained by other mathematicians. Then using these results, we bound all the zeros of p c (z) by limiting our consideration to different cases and we conclude that the location of the zeros of p c (z) depends on the value of c. The main theorems in this paper are Theorem 3.1 and Theorem 3.3. The structure of our paper is as follows. In section 2, we present some important preliminary results that will be used to prove our two main theorems. In section 3, we state and prove the main results of this paper. Theorem 3.1 determines an upper bound for the moduli of all the zeros of complex-valued harmonic polynomials and provides the zero inclusion regions for harmonic trinomial p c (z) = z n + cz k − 1 where where 1 ≤ k ≤ n − 1, n ≥ 3, c ∈ R + , and gcd(n, k) = 1. By considering different cases, Theorem 3.3 describes other zero inclusion regions, which is sharper, for the same trinomial p c (z) = z n + cz k − 1 where 0 < c < 1, c ≥ 1 and c ∈ C.

Preliminaries
In this section we review some important concepts and results that we will use later on to prove our theorems. We begin by stating the well known results such as Cauchy's bound, Descarte's Rule of Sign and some useful theorems and lemmas.
Theorem 2.1 (Cauchy's bound). Suppose p(x) = a n x n + a n−1 x n−1 + · · · + a 1 x+ a 0 be a polynomial of degree n with n ≥ 1. Cauchy's bound tells us where we can find the real zeros of a polynomial. For instance, the real zeros of the polynomial Then the number of positive real zeros of p(x) (counted with multiplicities) is either equal to the number of variations in sign in the sequence a 0 , · · · , a n of the coefficients or less than that by an even whole number. The number of negative zeros of p(x) (counted with multiplicities) is either equal to the number of variations in sign in the sequence of the coefficients of p(−x) or less than that by an even whole number.
Note that as the Fundamental Theorem of Algebra gives us an upper bound on the total number of roots of a polynomial, Descartes' Rule of Signs gives us an upper bound on the total number of positive ones. According to Descartes's Rule of Signs, polynomial p(x) has no more positive roots than it has sign changes and has no more negative roots than p(−x) has sign changes. A polynomial may not achieve the maximum allowable number of roots given by the Fundamental Theorem, and likewise it may not achieve the maximum allowable number of positive roots given by the Rule of Signs.
Proceeding from left to right, we see that the terms of the polynomial carry the signs + − +− for a total of three sign changes. Descartes' Rule of Signs tells us that this polynomial may have up to three positive roots and has no negative root. In fact, it has exactly three positive roots 1, 2, and 5.
Note that Theorem 2.4 is a classical solution in finding an upper bound for the moduli of all the zeros of the polynomial, which is analytic in general.
The following theorem is also another classical result for the location of the zeros of analytic complex polynomials that depends on algebraic equation's positive root. Descartes' Rule of Signs plays a great role in the proof as shown by Dehmer.
Theorem 2.5 (Dehmer, M. [2]). Let f (z) = a n z n +a n−1 z n−1 +· · ·+a 1 z+a 0 , a n = 0 be a complex polynomial. All zeros of f (z) lie in the closed disc K (0, max(1, δ)  Not only counting the number of zeros of analytic functions, Rouche's Theorem also plays a great role in bounding all the zeros of analytic functions, as we prove in the following lemma.
Proof. Note that R = 1 if and only if g(z) = z n . In this case, the assertion is obviously true. Therefore, we may assume that R > 1. If |z| = R, then using the Cauchy-Schwarz inequality Therefore, by Rouche's Theorem the zeros of g all lie in the open disk centered at the origin with radius R.

Main Results
Under this section, we bound all the zeros of complex-valued harmonic polynomials and specifically we bound the zeros of some families of trinomials to come up with a certain conclusion.

Bounds for the zeros of general harmonic polynomials
In this section we find an upper bound on the moduli of all the zeros of complexvalued harmonic polynomials. The following theorem derives a closed disk in which all zeros are included and hence it bounds the moduli of the zeros of the complex-valued harmonic polynomials. Proof. Observe that, since degg(z) = m, we have b m+1 = b m+2 = · · · = b n−1 = 0. Hence M = max |aj| |an| ∀j > m. Now by triangle inequality we have |b k ||z k | and the modulus of any complex number equals with modulus of its conjugate, after some algebraic manipulations we arrive at Then we get Descartes' Rule of Signs gives us that the function x n+1 − (1 + M )x n + M has exactly two distinct positive zeros, since x = 1 is a root with multiplicity one [17]. We then conclude that |p (  Proof. This corollary follows directly from Theorem (3.1) since it bounds all the zeros of any complex-valued harmonic polynomials. Now This shows that |p c (z)| > 0 for |z| > R. Hence the result.

Sharper bounds for the zeros of harmonic trinomials
In the previous section, we have found the upper bound on the moduli of all the zeros of complex-valued harmonic polynomials stated in Theorem 3.1. In the following theorem we find upper and lower bounds for the moduli of the zeros of harmonic trinoimial p c (z) = z n + cz k − 1 by considering different cases. Theorem 3.3. Let p c (z) = z n + cz k − 1 where 1 ≤ k ≤ n − 1, n ≥ 3, and gcd(n, k) = 1. Then the zeros of p c (z) lies in the annular region n−k if z is a zero of p c (z) with |z| ≥ 1 and c ∈ C.
Note that in accordance with work of Anderson et al [19] and Wang [17] both trinomial equations in lemma (3.4) have only one positive real root. respectively, then the roots of z n + az k + b = 0, lies in the ring y 1 < |z| < y 2 where ab = 0, 1 ≤ k ≤ n − 1, n ≥ 3, and gcd(n, k) = 1.
Note that the roots of trinomial equation z n + az k + b = 0 and the roots of its harmonic equivalent are bounded in the same ring with respect to their modulus. In particular, since p c (z) = z n + cz k − 1 = 0 is a harmonic equivalent to q(z) = z n + az k + b = 0, the zero inclusion regions of p and q are the same. Combining lemma (3.6) and lemma (3.7) plays a crucial role to prove Theorem 3.3.
Proof. It was shown in [15] that p c (z) has n distinct zeros. From p c (z) = z n + cz k − 1, we have by triangle inequality |z n + cz k | = 1 ≤ |z| n + c|z| k = |z| n + c|z| k . Now then consider lemma (3.4). By taking the values a = c and b = −1, the forward substitution helps us to finish the desired proof. Lemma (3.5) also helps us to conclude the following result. Here by following the same procedures, we show that β 1 < α 1 and β 2 > α 2 .
Lemma 3.7. The root of harmonic trinomial p c (z) = z n + cz k − 1 lies in the ring β 1 < |z| < β 2 where β 1 and β 2 are the positive real roots of the equations x n−k +c−1 = 0 and x n−k − c − 1 = 0 respectively.
Now we are in position to prove our theorem, Theorem 3.3.

Conclusion
In this paper we have found an upper bound for the moduli of all the zeros of complex-valued harmonic polynomials. We derived here a closed disk in which all zeros are included and we bound the moduli of the zeros of the complex-valued harmonic polynomials in general. More specifically, it has been shown that the number of zeros of p c (z) = z n + cz k − 1 changes as c varies by Brilleslyper et al [15]. In this paper we have derived a locations for the zeros of p c (z) and the result shows that the location of the zeros also changes as c varies.