Cosine polynomials with few zeros
نویسندگان
چکیده
In a celebrated paper, Borwein, Erdélyi, Ferguson and Lockhart constructed cosine polynomials of the form f A ( x ) = ∑ ∈ cos , with ⊆ N | n as few 5 / 6 + o 1 zeros in [ 0 2 π ] thereby disproving an old conjecture Littlewood. Here we give sharp analysis their constructions and, result, prove that there exist examples C log 3 zeros.
منابع مشابه
Some compact generalization of inequalities for polynomials with prescribed zeros
Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$, $k^2 leq rRleq R^2$ and for $Rleq r leq k$. Our results refine and generalize certain well-known polynomial inequalities.
متن کاملRandom polynomials having few or no real zeros
Consider a polynomial of large degree n whose coefficients are independent, identically distributed, non-degenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros with probability n−b+o(1) as n → ∞ through integers of the same parity as the fixed integer k ≥ 0. In particular, the probability that a random polynomial of...
متن کاملComputing Polynomials with Few Multiplications
It is a folklore result in arithmetic complexity that the number of multiplication gates required to compute a worst-case n-variate polynomial of degree d is at least
متن کاملGeneric Polynomials with Few Parameters
We call a polynomial g(t1, . . . , tm, X) over a field K generic for a group G if it has Galois group G as a polynomial in X, and if every Galois field extension N/L with K ⊆ L and Gal(N/L) ≤ G arises as the splitting field of a suitable specialization g(λ1, . . . , λm, X) with λi ∈ L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic p...
متن کاملOn the Zeros of Cosine Polynomials: Solution to a Problem of Littlewood
Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [12, problem 22] poses the following research problem, which appears to still be open: Problem. “If the nj are integral and all different, what is the lower bound on the number of real zeros of PN j=1 cos(njθ)? Possibly N −1, or not much less.” No progress appears to have been made on this in the last half century. We...
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: Bulletin of The London Mathematical Society
سال: 2021
ISSN: ['1469-2120', '0024-6093']
DOI: https://doi.org/10.1112/blms.12468