Dragging the roots of a polynomial to the unit circle

Abstract

Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials p α to each such polynomial p, and define (p), (p) to be the sharp threshold values of α that guarantee that, for all larger values of the parameter, p α has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree. Interlacing implies circle rootedness, hence (p) ≥ (p), and this inequality is often used for showing circle rootedness. Both (p) and (p) turn out to be semi-algebraic functions of the coefficients of p, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of (p) leads to a rich classification of real selfinversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which (p) = (p), whereas in general the quotient (p) (p) is shown to be unbounded as the degree grows. Several examples and open questions are presented.

Large finite products of small fractions

“…The motivating example in [3] is D n (5, 3, 4, π/2, π/2; sin); some routine algebraic manipulation show that ε = π 2 conforms to the requirements of the Theorem, and we obtain D n ∼ C √ n for some constant C. The result will be obtained by comparing D n and K n . The asymptotics for K n is well known (it essentially appears in [1, 11 th formula line]).…”

“…where the terms go on while the arguments of sin stay below π 2 . The proof of an important result in Mandel and Robins [3] hinges on showing that D n grows unboundedly with n. It is shown there that D n = Ω(n 1 2 −ε ), which is enough; here we remove the annoying −ε from the exponent and determine the precise order of growth.…”

An Algebraic Approach to Degenerate Bernoulli Numbers

“…In this section, we show that the degenerate Bernoulli polynomial βk (m, x) has palindromic properties if m is odd. A consequence of this is that the degenerate Bernoulli numbers vanish for certain k depending on m. Definition 6.1 (Palindromic Polynomial [12,18]). Let p(x) = α r x r + • • • + α s x s be a non-zero polynomial, with real coefficents and α r , α s = 0.…”