On the absolute length of polynomials having all zeros in a sector

Abstract

Let α be an algebraic integer whose all conjugates lie in a sector | arg z| ≤ θ, 0 ≤ θ < 90 •. Using the method of auxiliary functions, we first improve the known lower bounds of the absolute length of totally positive algebraic integers, i.e., when θ is equal to 0. Then, for 0 < θ < 90 • , we compute the greatest lower bound c(θ) of the absolute length of α, for θ belonging to eight subintervals of [0, 90 •). Moreover, we have a complete subinterval, i.e., an interval on which the function c(θ) describing the minimum on the sector | arg(z)| ≤ θ is constant, with jump discontinuities at each end. Finally, we obtain an upper bound for the integer transfinite diameter of the interval [0, 1] from the lower bound of the absolute length. The polynomials involved in the auxiliary functions are found by our recursive algorithm.

The Absolute -Measure of Totally Positive Algebraic Integers

“…, c J ), the c j are positive real numbers, the Q j are nonzero polynomials in Z[x] and I is a real interval. For instance, if ψ(x) = log(x + 1), the auxiliary function (2.1) can be applied for the lower bound of the absolute length R(α) = L(α) 1/d , where L(α) = d i=0 |a i | (for more details, see [5,6,14]); and ψ(x) = log(max{1, x}) for the lower bound of the absolute Mahler measure Ω(α) = M(α) 1/d , where M(α) = |a 0 | d i=1 max(1, |α i |) (for more details, see [5,7,14]). To prove Theorem 1, we take ψ(x) = x k and I = (0, +∞) (see [9,13,15,24]) for each integer in the range 2 ≤ k ≤ 15.…”

Errata and Addenda to Mathematical Constants

“…Integer Chebyshev Constant. The bounds 0.4213 < χ(0, 1) < 0.422685 are currently best known [552,553,554,555]. Other values of χ(a, b) and various techniques are studied in [556].…”

“…(his actual lower bound 1.5377 used χ(0, 1) < 0.42291334 from [554]; we use the refined estimate from [555]). A follow-up essay on real transfinite diameter is [562].…”

An Analogue of the Schur–siegel–smyth Trace Problem

“…These improvements came from our recursive algorithm, based on Wu's algorithm but where the polynomials are found by induction. We applied this method to measures such as the trace [2], the length [3] and the house [4], as well as to unusual measures (see [6][7][8]). Here we prove the following result.…”