The Mahler Measure of Polynomials with Odd Coefficients
نویسندگان
چکیده
The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with ±1 coefficients and degree at most 72 are also determined.
منابع مشابه
Lehmer ’ s problem for polynomials with odd coefficients
We prove that if f(x) = ∑n−1 k=0 akx k is a polynomial with no cyclotomic factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k < n, then Mahler’s measure of f satisfies log M(f) ≥ log 5 4 ( 1 − 1 n ) . This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n−1, and at least one noncyclotomic fact...
متن کاملEXTREMAL MAHLER MEASURES AND Ls NORMS OF POLYNOMIALS RELATED TO BARKER SEQUENCES
In the present paper, we study the class LPn which consists of Laurent polynomials P (z) = (n+ 1) + n ∑ k=1 k – odd ck(z k + z−k), with all coefficients ck equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length — binary sequences with minimal possible autocorrelations. By using an elementary (but not trivial) analytic argument, we prove that polynomials Rn(z) w...
متن کاملLower Bounds for the Mahler Measure of Polynomials on Subarcs
We give lower bounds for the Mahler measure of polynomials with constrained coefficients, including Littlewood polynomials, on subarcs of the unit circle of the complex plane. This is then applied to give an essentially sharp lower bound for the Mahler measure of the Fekete polynomials on subarcs.
متن کاملBarker Sequences and Flat Polynomials
A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding the existence of polynomials with ±1 coefficients that remain flat over the unit circle according to some cr...
متن کاملPolynomials with small Mahler measure
We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3, test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004