Zeros of sections of exponential sums

Abstract

Abstract. We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the n-th section of the exponential sum into "genuine zeros", which approach, as n → ∞, the zeros of the exponential sum, and "spurious zeros", which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a "rosette", a finite collection of curves on the complex plane, resembling the rosette. We derive also the large n asymptotics of the "transitional zeros", the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szegö about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.

What is the probability that a large random matrix has no real eigenvalues?

“…The statement of Lemma 4, where cosh n pnxq is replaced by cosh n´1 pnxq, is easily deduced from equation (65). Our proof builds on the ideas of [4] dedicated to the study of sections of exponential series (Taylor polynomials generated by exp). Let e n be a section of exponential series defined by e n pxq " n ÿ j"0 x j j!…”

Extremal laws for the real Ginibre ensemble

“…Lemma 3.3 [Bleher and Mallison (2006)]. For small enough δ > 0, let now µ(z) = √ z − log z − 1 be uniquely defined as analytic in |z − 1| < δ with µ(1 + x) > 0 for 0 < x < δ.…”

“…Both Bleher and Mallison (2006) and Kriecherbauer et al (2008) contain more detailed and complete asymptotics along the lines stated in Lemmas 3.3 and 3.4; we record only what is used here. Also, as is easy to check, the appraisals of all three lemmas apply without change to e n−2 (rather than say e n , e n−1 ) for n large enough.…”

Appell polynomials and their zero attractors

“…(Left-Half Plane) Let 1/3 < α < 1/2 and 1 ≤ j. On any compact subset K of {w : ℜw < 1}, we have (1) S n−1 (nw) e nw = 1 − (we 1−w ) where the big O constant holds uniformly for x ∈ K and D w is the usual differential operator.…”

“…For both parts, we can use the asymptotics given in Theorem 5.18. For part (1), let f n (x) = √ 2πng(1/x)p n (nx)/(xe) n . Let a be a dominant zero of g, and let C be a proper subcurve of ∂D 0 ∩ ∂D ± a .…”