On Nonnegative Cosi Ne Polynomials with Nonnegative, Integral Coeecients
نویسنده
چکیده
We show that there is p 0 > 0 and p 1 ; : : :; p N non-negative integers, such that 0 p(x) = p 0 + p 1 cos x + + p N cos Nx and p 0 s 1=3 for s = P N j=0 p j , improving on a result of Odlyzko who showed the existence of such a polynomial p that satisses p 0 (s log s) 1=3. Our result implies an improvement of the best known estimate for a problem of Erdd os and Szekeres. As our method is non-constructive, we also give a method for constructing an innnite family of such polynomials, given one good \seed" polynomial.
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