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 کاملThe Mahler Measure of Polynomials with Odd Coefficients
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.
متن کاملHeights of roots of polynomials with odd coefficients
We show that the height of a non-zero non root of unity α which is the zero of a polynomial with all odd coefficients of degree n satisfies h(α) ≥ 0.4278 n+ 1 . More generally we obtain bounds when the coefficients are all congruent to 1 modulo m for some m ≥ 2.
متن کاملPolynomials À La Lehmers and Wilf
We show that a period polynomial introduced by the Lehmers coincides with a generalized Wilf polynomial. Résumé. Nous montrons qu’un polynôme période introduit par les Lehmer cöıncide avec un polynôme de Wilf généralisé.
متن کاملOn a problem of Byrnes concerning polynomials with restricted coefficients
We consider a question of Byrnes concerning the minimal degree n of a polynomial with all coefficients in {−1, 1} which has a zero of a given order m at x = 1. For m ≤ 5, we prove his conjecture that the monic polynomial of this type of minimal degree is given by ∏m−1 k=0 (x 2 − 1), but we disprove this for m ≥ 6. We prove that a polynomial of this type must have n ≥ e √ m(1+o(1)), which is in ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2007
ISSN: 0003-486X
DOI: 10.4007/annals.2007.166.347