The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

Abstract:

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets E_k⊂ℝ of admissible coefficients, we construct a polynomial 

P_n(x)=1+∑_{k=1}^n ε_kx^k, 

ε_k∈E_k, with at least c√n distinct roots in [0,1], which matches the classical upper bound up to the value of the constant c>0. Our sufficient conditions cover the Littlewood (E_k={−1,1}) and Newman (E_k={0,(−1)^k}) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence E_k is periodic.

Joint with Markus Jacob.