On the number of roots for harmonic trinomials

Barrera, Gerardo; Barrera, Waldemar; Navarrete, Juan Pablo

doi:10.1016/j.jmaa.2022.126313

Cited by 6 publications

(6 citation statements)

References 51 publications

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“…In addition, Theorem 1.2 with the help of the following results in [1], Lemma 2.6, Lemma 2.11, Lemma A.3 and Proposition 2.3, yields the following statements.…”

Section: Introduction Main Results and Their Consequencesmentioning

confidence: 87%

“…By Bézout's Theorem ([52, Theorems 1, 5]) it follows that (1.2) has at most (n + m) 2 roots. Recently, in [1,Corollary 1.4], it is shown that (1.2) has at most n + 3m roots. Moreover, such bound is sharp in the sense that there exist harmonic trinomials with exactly n + 3m roots.…”

Section: Introduction Main Results and Their Consequencesmentioning

confidence: 99%

“…We assume that n and m are co-prime numbers. Since |A|r n+m , |B|r m and |C| are the side lengths of some degenerate triangle, the contrapositive of [1,Lemma 2.11] applied to h(z) := e −iγ h(z), z ∈ C yields…”

Section: Then We Havementioning

confidence: 99%

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Egerváry's theorems for harmonic trinomials

Barrera,

Navarrete

2024

Acta Math. Hungar.

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We study the arrangements of the roots in the complex plane for the lacunary harmonic polynomials called harmonic trinomials. We provide necessary and sufficient conditions so that two general harmonic trinomials have the same set of roots up to a rotation around the origin in the complex plane, a reflection over the real axis, or a composition of the previous both transformations. This extends the results of Jenő Egerváry given in [19] for the setting of trinomials to the setting of harmonic trinomials.

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“…In addition, Theorem 1.2 with the help of the following results in [1], Lemma 2.6, Lemma 2.11, Lemma A.3 and Proposition 2.3, yields the following statements.…”

Section: Introduction Main Results and Their Consequencesmentioning

confidence: 87%

Section: Introduction Main Results and Their Consequencesmentioning

confidence: 99%

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Egerváry's theorems for harmonic trinomials

Barrera,

Navarrete

2024

Acta Math. Hungar.

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“…We now consider the complex dilatation of the second family (8). We again desire this dilatation to be of the form given in (9).…”

Section: The Subfamiliesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Zeros of Convex Combinations of Elementary Families of Harmonic Functions

Brooks,

Dixon,

Dorff

et al. 2023

Mathematics

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Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the number of zeros of convex combinations of members of these families and show that it is possible for a convex combination of two members of a family to have more zeros than either of its constituent parts. Our main tool to prove these results is the harmonic analog of Rouché’s theorem.

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