A New Lower Bound for the Maximal Valence of Harmonic Polynomials
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A New Lower Bound for the Maximal Valence of Harmonic Polynomials Seung-Yeop Lee1 · Andres Saez1
Received: 31 December 2015 / Revised: 22 May 2016 / Accepted: 31 May 2016 / Published online: 2 August 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract We find a new lower bound for the maximal number of zeros of harmonic polynomials, p(z) + q(z), when deg p = n and deg q = n − 2. Keywords Harmonic polynomial · Valence · Large degree asymptotics Mathematics Subject Classification 30C55 · 30C10
1 Introduction and Result Given two polynomials p(z) and q(z) of degrees n and m respectively, the maximal number of roots (i.e. maximal valence) of the harmonic polynomial, p(z) + q(z), is not known [11] except for a few cases (e.g. when m = n − 1 [13,14] and when m = 1 [9,10]). See also [2,5–8,12]. Recently there have been several results [1,3,4] on the lower bounds of the maximal valence, see Table 1. In this paper we suggest a new lower bound of the maximal valence when (deg p, deg q) = (n, n−2), by studying specific harmonic polynomials defined below. Given a positive integer n let us define two polynomials, p(z) = S(z) + T (z) and q(z) = S(z) − T (z), where n−1 z − (n − 1) . S(z) = i z n , T (z) = i z + 1
(1)
Communicated by Stephan Ruscheweyh.
B 1
Seung-Yeop Lee [email protected] Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Ave, CMC 342, Tampa, FL 33620, USA
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S.-Y. Lee, A.Saez
Table 1 The known maximal valence of p + q (deg p, deg q)
(n, m)
(n, n − 1)
(n, n − 3)
(n, 1)
Maximal valence
≥m 2 + m + n
n2
≥n 2 − 3n + O(1)
3n − 2
It follows that deg p = n and deg q = n − 2. Since the maximal valence for (n, m) = (3, 1) is known (see the above table), we only consider n ≥ 4 in this paper. Theorem Given n ≥ 4, let the polynomials p and q be given as above. Let kmax (n) be defined by πk 2k − 1 π − n cot kmax (n) = max k : (n − 2) cot >0 . 1≤k≤n/2 2n − 4 n Then the total number of zeros, counting their multiplicities, of p(z) + q(z) is given by n 2 − 2n + 2 + 4kmax (n). The asymptotic behavior of kmax (n) as n → ∞ is given by kmax (n) =
X 1 − 4 2π
n + O (1) ≈ 0.13237n + O (1)
where X ≈ 0.73908513321516 is the unique solution to the equation X = cos X (Table 2). Remark 1 For general n and m, there exists a conjecture by Wilmshurst [13] on the largest valence of the harmonic polynomials. Though the conjecture has been disproved [3,4] for a number of cases, it has not been checked for many other cases including the case considered in this paper. Our theorem says that the maximal valence is greater at least by 4kmax (n) − 2 ≈ 0.52948n + O(1) as n → ∞ than the conjectured value of n 2 − 2n + 4. Our theorem also improves upon the more recent conjecture by the authors that suggests n 2 − 3n/2 + O(1) for the asymptotic maximal valence as n grows to infinity. In fact, the current project is motivated by the latter conjecture, which is no longer valid according to our theorem. Remark 2 The specific harmonic polynomials that we consider in this pap
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