Egerv\'ary's theorems for harmonic trinomials

In this manuscript, 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 J. Egerv\'ary 1930 for the setting of trinomials to the setting of harmonic trinomials.

The computation and the quantitative location of the roots for polynomials are important in many research areas, and therefore a vast literature in both pure mathematics and applied mathematics has been produced, we refer to [6,40,44,45,46,47] and the references therein.
Given two positive integers m and n, a trinomial of degree n + m is a lacunary polynomial with three terms of the form (1.1) T (z) := Az n+m + Bz m + C for all z ∈ C, where A, B and C are non-zero complex numbers.Despite the apparent simplicity of (1.1), the well-known works of P. Ruffini, N.H.Abel and É. Galois imply that for n + m ≥ 5 and generic trinomials of the form (1.1) there is no formula for their roots in terms of the so-called radicals.For the literature reporting geometric, topological, quantitative and qualitative behavior of the roots for trinomials of the form (1.1) we refer to [2,3,4,5,7,8,9,10,14,15,16,17,18,19,20,27,32,33,41,42,43,48,49] and the references therein.
In [19] J. Egerváry analyzes the roots of general trinomials.More precisely, he studied (I) the arrangements of the roots of (1.1) in the complex plane, that is, provides necessary and sufficient conditions so that two general trinomials possess 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.The latter is an equivalence relation, which in the sequel we refer to as its Egerváry equivalent, see Definition 1.1 below.(II) the description of geometric sectors for the localization of the roots of (1.1).
Since [19] is written in Hungarian, many of the results given there have been rediscovered afterwards.We refer to [48] for an English review of [19].
In this manuscript, we extend (I) to the setting of harmonic trinomials.More precisely, to the setting of lacunary harmonic polynomials of the form ( (e) A geometric degenerate triangle condition on the modulus of the coefficients of (1.2) so that (1.2) is Egerváry equivalent to a harmonic trinomial with real coefficients, see Theorem 1.8 below.(f) Two harmonic trinomials of the form (1.2) with roots having the same complex modulus (such roots may be different) and satisfying that the ratio between the complex modulus of their respective coefficients with the same degree is constant, are Egerváry equivalent, see Theorem 1.9 below.By Bezout's Theorem (Theorem 1 and Theorem 5 in [52]) it follows that (1.2) has at most (n + m) 2 roots.Recently, in Corollary 1.4 of [1], 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.In general, there exist harmonic polynomials with exactly (n + m) 2 roots, see for instance Section 2 in [52] or p. 2080 of [12].

Preliminaries and main results.
In this subsection, we present the preliminaries and state the results of this manuscript.
Given m, n ∈ N := {1, 2, . . ., }, we consider two harmonic trinomials We start with the following definition, which rigorously encodes the arrangements of roots that are equivalent.

Definition 1.1 (Egerváry equivalent).
Let h 1 and h 2 be the harmonic polynomials given in (1.3) and (1.4), respectively.We say h 1 and h 2 are Egerváry equivalent if and only if the set of roots of h 1 differs of the set of roots of h 2 by (a) a rotation around the origin in the complex plane, (b) a reflection over the real axis, (c) a composition of both transformations given in (a) and (b).More precisely, there exist a non-zero complex number c and a real number δ satisfying (1.5) where i denotes the imaginary unit and We note that Definition 1.1 defines an equivalence relation.In addition, we observe that (1.5) and (1.6) are mutually exclusive whenever some of the coefficients A 2 , B 2 or C 2 is not a real number.Indeed, if (1.5) and (1.6) both hold true, we have h 1 (z) = ch 2 (e iδ z) = ch 2 (e iδ z) for all z ∈ C, which yields The latter implies A 2 , B 2 and C 2 are real numbers.
Along this manuscript, |ζ| denotes the complex modulus of the given complex number ζ.We recall that the polar representation of ζ is given by ζ = |ζ|e iϕ , where ϕ ∈ [0, 2π) is the argument of ζ.Moreover, for any real numbers x and y, we write that x ≡ y mod 2π if and only if x − y = 2kπ for some k ∈ Z.The first main result of this manuscript is the following extension of the results given in Equation (2) and Equation (3) in p. 37 of [19] or Theorem 1 in the survey [48], to the setting of harmonic trinomials.It reads as follows.

Theorem 1.2 (Egerváry's Theorem for harmonic trinomials).
Let h 1 and h 2 be the harmonic polynomials given in (1.3) and (1.4), respectively.Then the following holds true: h 1 and h 2 are Egerváry equivalent if and only if where α 1 , α 2 , β 1 , β 2 , γ 1 and γ 2 are the arguments in the polar representation of The proof is given in Subsection 2.1.

Remark 1.3 (About the choice of ±).
We point out that (1.8) reads The following corollary is the analogous of the results given in Equation ( 1), Equation ( 5) and Equation ( 6) in p. 38 of [19] or p. 100 in [48] to the setting of harmonic trinomials.It reads as follows.

Corollary 1.4 (The class of harmonic trinomials with real coefficients).
Let h(z) = Az n+m +Bz m +C for all z ∈ C be a harmonic trinomial whose coefficients A, B and C are non-zero complex numbers.We consider the polar representation of A, B and C, that is, A = |A|e iα , B = |B|e iβ , C = |C|e iγ .Then the following statements are equivalent.
(i) The harmonic trinomial h is Egerváry equivalent to a harmonic trinomial with real coefficients.(ii) The angular relation is a real number.
The proof is provided in Subsection 2.2.
The following corollary is the analogous of Statement III in p. 40 of [19].

Corollary 1.5 (Different roots with the equal modulus).
Let h(z) = Az n+m +Bz m +C for all z ∈ C be a harmonic trinomial whose coefficients A, B and C are non-zero complex numbers.We consider the polar representation of A, B and C, that is, A = |A|e iα , B = |B|e iβ , C = |C|e iγ .Assume that h has at least two different roots with the same modulus.Then h is Egerváry equivalent to a harmonic trinomial with real coefficients.
The proof is presented in Subsection 2.3.
In the sequel, we remark that the converse of the Statement III in p. 40 of [19] does not hold true in general.For n + m ∈ N \ {1, 2} we claim that all the roots of h(z) = Az n+m + Bz m + C for all z ∈ C, where A, B, C ∈ R \ {0} cannot be real numbers.Indeed, by Descartes' rule of signs we have that h has at most two real roots and hence h has at least one complex root ζ.It is not hard to see that ζ is also a root of h and hence the converse of Corollary 1.5 holds true.
The following corollary is the analogous of Statement IV in p. 40 of [19].

Corollary 1.7 (Root with multiplicity at least two).
Let h(z) = Az n+m +Bz m +C for all z ∈ C be a harmonic trinomial whose coefficients A, B and C are non-zero complex numbers.We consider the polar representation of A, B and C, that is, A = |A|e iα , B = |B|e iβ , C = |C|e iγ .Assume that h has a root of multiplicity at least two with modulus r.Then h is Egerváry equivalent to a harmonic trinomial with real coefficients.Moreover, The proof is provided in Subsection 2.4.The analogous of the following theorems are not given in [19].We state them here since they are interesting on their own.They are deduced using Theorem 1.2 together with the following results in [1], Lemma 2.6, Lemma 2.11, Lemma A.3 and Proposition 2.3.

Theorem 1.8 (Geometric degenerate condition).
Let h(z) = Az n+m +Bz m +C for all z ∈ C be a harmonic trinomial whose coefficients A, B and C are non-zero complex numbers.We consider the polar representation of A, B and C, that is, A = |A|e iα , B = |B|e iβ , C = |C|e iγ .Assume that there exists a root of h with modulus r such that |A|r n+m , |B|r m and |C| are the side lengths of some degenerate triangle.Then h is Egerváry equivalent to a harmonic trinomial with real coefficients of the form The proof is given in Subsection 2.5.
Theorem 1.9 (Common root with the same modulus).
Let h 1 and h 2 be the harmonic polynomials given in (1.3) and (1.4), respectively.Assume that (1.7) holds true.In addition, assume that there exists ζ 1 and ζ 2 roots of h 1 and h 2 , respectively, and satisfying In particular, h 1 and h 2 are Egerváry equivalent.
The proof is provided in Subsection 2.6.The rest of the manuscript is organized as follows.In Section 2 we provide the proofs of the results given in Section 1.More precisely, in Subsection 2.1 we show Theorem 1.2, in Subsection 2.2 we provide the proof of Corollary 1.4, in Subsection 2.3 we give the proof of Corollary 1.5, in Subsection 2.4 we provide the proof of Corollary 1.7, in Subsection 2.5 we show Theorem 1.8 and in Subsection 2.6 we prove Theorem 1.9.Finally, in Appendix A we state auxiliary results that have been used throughout the manuscript.

Proofs of the results
In this section, we present the proofs of the results stated in Subsection 1.2.

Proof of Theorem 1.2.
Assume that h 1 and h 2 are Egerváry equivalent.By Definition 1.1 it is not hard to see that there exist a non-zero complex number c and a real number δ satisfying (2.1)
In the sequel, we assume that (1.7) and (1.8) are valid.In particular, we have Without loss of generality, we can assume that r = 1.Since r = 1, (2.7) implies the existence of real numbers θ 1 , θ 2 and θ 3 such that (2.9) Then we have (2.10) By (2.8) and (2.10) we obtain By Lemma A.1 in Appendix A we have that the solutions of (2.11) are parametrized as follows (2.12) for some δ ∈ R. By (2.9) and (2.12) we obtain for all z ∈ C. In the case of the proof is analogous and we omit it.

Proof of Corollary 1.5.
Assume that h has two different roots with modulus r > 0. After a rotation, one can see that h is Egerváry equivalent to a harmonic trinomial h with roots ζ 1 = r and ζ 2 = re iθ , where θ ∈ (0, 2π).Without loss of generality, we assume that h = h.Since h(ζ 1 ) = h(ζ 2 ) = 0, we have Recall the identity for any t ∈ R.
Then we have which with the help of (iii) of Corollary 1.4 yields the statement.
After a rotation, without loss of generality, we can assume that h has a real root r > 0 with multiplicity at least two.The function γ(x) := Ax n+m + Bx m + C, x ∈ R represents a curve in the complex plane C. Since r is a root of h with multiplicity at least two and h(x) = γ(x) for all x ∈ R, we have γ(r) = γ ′ (r) = 0, where γ ′ denotes the derivative of γ.The latter reads as follows which with the help of (iii) of Corollary 1.4 yields the statement.
2.5.Proof of Theorem 1.8.By (ii) of Corollary 1.4, it is enough to show (1.9).By hypothesis we have that h(r) = 0. 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 Lemma 2.11 in [1] applied to h(z) := e −iγ h(z), z ∈ C yields The latter implies (1.9).Moreover, by (2.16) we have that h is Egerváry equivalent to g u,v for some u, v ∈ {−1, 1}, where g u,v is defined in (1.10).
We continue with the proof when d := gcd(n + m, m) ∈ {2, . . ., m}.Observe that gcd(n, m) = d.Let n ′ := n/d and m ′ := m/d and note that gcd(n ′ , m ′ ) = 1.Since h has a root of modulus r, we have that the harmonic trinomial has a root of modulus r d .Then the previous discussion for the co-prime case implies Multiplying by d in both sides the preceding inequality yields In addition, H is Egerváry equivalent to for some u, v ∈ {−1, 1}.The change of variable z → z d yields that h is Egerváry equivalent to g u,v for some u, v ∈ {−1, 1}.
In the sequel, we show (a).We start with the following observation.The relation |C| = |A|r n+m + |B|r m holds true if and only if g −1,−1 (r) = 0.By Descartes' rule of signs we have that r is the unique positive real number satisfying g −1,−1 (r) = 0.By Theorem 1.2 one can verify that (2.17) if and only if n is an even number, g 1,−1 if and only if m is an even number, where g u,v is defined in (1.10).Now, we assume that n and m are co-prime numbers.Then we claim that g −1,1 , g 1,1 and g 1,−1 are never Egerváry equivalent between them.Indeed, we start assuming that n + m is an even number.By (2.17) we have g −1,−1 is Egerváry equivalent to g −1,1 .Since n and m are co-prime numbers, the assumption that n + m is an even number imply that n and m are odd numbers.Recall that being Egerváry equivalent is an equivalence relation.Hence (2.17) yields that g −1,1 cannot be Egerváry equivalent to g 1,1 neither g 1,−1 when n + m is an even number.
We now claim that g 1,1 and g 1,−1 do not have a root of modulus r.We start showing that g 1,−1 does not have a root of modulus r.Indeed, by contradiction assume that there exists ζ = re iθ with θ ∈ [0, 2π) such that g 1,−1 (ζ) = 0, that is, Hence, (2k + 1)m = 2k ′ (n + m), which is a contradiction since m and 2k + 1 are odd numbers.Now, we prove that g 1,1 has no root of modulus r.By contradiction, assume that there exists a root of g 1,1 of the form ζ = re iθ with θ ∈ [0, 2π).Then it follows that |A|r n+m e i(θ(n+m)) + |B|r m e −i(θm) + |C| = 0. Similarly to the previous case, we obtain θ(n + m) ≡ π mod 2π and − θm ≡ π mod 2π, which implies (2k + 1)m = (2k ′ + 1)(n + m) for some k, k ′ ∈ Z.This yields a contradiction since m and 2k + 1 are odd numbers and n + m is an even number.We observe that (1.10) yields that h is Egerváry equivalent to g u,v for some u, v ∈ {−1, 1}.The preceding analysis implies that h is Egerváry equivalent to g −1,−1 , which is also Egerváry equivalent to g −1,1 .
The proof when n is an even number and the proof when m is an even number follow similarly and we omit them.In summary, the proof of (a) is complete.Moreover, the proofs of (b) and (c) are analogous.
Since (1.7) is valid, without loss of generality we assume that We now assume that n and m are co-prime numbers.Case (1).Assume that |A 1 |r n+m , |B 1 |r m and |C 1 | are not the side lengths of any triangle.By Lemma 2.6 in [1] we have that there is no root of modulus r for the harmonic trinomial h 1 and h 2 , which yields a contradiction.Case (2) where w 1 and w 2 are the angles opposite to the side lengths |A 1 |r n+m and |A 2 |r m , respectively.We note that ω * ,1 (r) = ω * ,2 (r).By Proposition 2.3 in [1] for each j = 1, 2 we have that P * ,j + ω * ,j or P * ,j − ω * ,j are integers numbers.If P * ,1 + ω * ,1 (r) and P * ,2 + ω * ,2 (r) are integers, then P * ,1 − P * ,2 is an integer and by (2.18) we deduce The remainder cases are similar and hence we omit their proofs.Case Proof.The proof follows step by step from the proof of Lemma A.1 replacing 2π by π.

Remark 1 . 6 ( 3 6 z − 1 4 1
Converse of Corollary 1.5 for degree three or more).Let h(z) = z 2 + √ for all z ∈ C. By Corollary 1.4 in [1] we have that h has at most four different roots in C. In fact, a straightforward computation yields that h has only two roots, which are given by z Since |z 1 | = |z 2 |, we have that the converse of Corollary 1.5 is not valid when n + m = 2, i.e., n = m = 1.
1.2) H(z) := Az n+m + Bz m + C for all z ∈ C, where A, B and C are non-zero complex numbers, and ζ denotes the complex conjugate of the given complex number ζ, see Theorem 1.2 below.As a consequence of Theorem 1.2 we obtain the following results.(a) A characterization of the class of harmonic trinomials of the form (1.2) which are Egerváry equivalent with a harmonic trinomial with real coefficients, see Corollary 1.4 below.(b) A harmonic trinomial of the form (1.2) with different roots having the same complex modulus is Egerváry equivalent to a harmonic trinomial with real coefficients, see Corollary 1.5 below.(c) A harmonic trinomial of the form (1.2) with a root of multiplicity at least two is Egerváry equivalent to a harmonic trinomial with real coefficients, see Corollary 1.7 below.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.
where A 1 , B 1 and C 1 are real numbers.In particular,(2.13)α 1 ≡ 0 mod π, β 1 ≡ 0 mod π and γ 1 ≡ 0 mod π, where α 1 , β 1 and γ 1 are the arguments A 1 , B 1 and C 1 , respectively.By Theorem 1.2 (applied to h and g) we have r > 0 be fixed.Then the following straightforward remark is true: |A 1 |r n+m , |B 1 |r m and |C 1 | are the side lengths of a triangle ∆ 1 (it may be degenerate), if and only if, |A 2 |r n+m , |B 2 |r m and |C 2 | are the side lengths of a triangle ∆ 2 .In fact, ∆ 1 and ∆ 2 are congruent.By hypothesis, h 1 and h 2 have roots (such roots may be different) of modulus r for some r > 0. The proof is divided in three cases accordingly to |A 1 |r n+m , |B 1 |r m and |C 1 | are the side lengths of some triangle.