Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials

Real roots of random polynomials: expectation and repulsion

“…The truly remarkable fact about (6) is that the error term EN n,N (0,1) − 2 π log n tends to a limit as n tends to infinity. The question here is: Is this a universal phenomenon, which holds for general random polynomials, or a special property of the Gaussian one ?…”

“…It is clear that the computation leading to (6) is not applicable for general random polynomials, as the explicit formula in (3) is available only in the Gaussian case, thanks to the unitary invariance property of this particular distribution. For many natural variables, such as Bernoulli, there is little hope that such an explicit formula actually exists.…”

On Newman and Littlewood polynomials with prescribed number of zeros inside the unit disk

“…For instance, the angular equidistribution property [2,45,50,56,81,93] of zeros, the range of absolute values of zeros compared to their argument angle or zero multiplicity [4,5,106]. Asymptotic bounds for the number of real zeros [16,27,44,109,110], their expected values [74,75,49] are available; as well as the concentration inequalities for the probability to have a zero at a fixed point [76]; also asymptotic estimates on the number of zeros in circular and polygonal regions [13,17,43]; bounds [1] for the maximal vanishing multiplicity at roots of unity (especially at point z = 1, see [18,25,26,35,36]), etc.…”

“…Over last 15 years, many results have been established on U (f ) for polynomials f ∈ N and L. The first essential fact, that U (f ) ≥ 1 holds for all self-reciprocal {0, 1} and {−1, 1} polynomials, was proved (using different methods) by Konyagin and Lev [68], Kovalina and Matache [67], Mercer [78] and Erdélyi [44]. For self-reciprocal f ∈ N ∪ L of odd degree ≥ 3, Mukunda [85] raised the lower bound to U (f ) ≥ 3.…”

“…Almost simultaneously when Erdélyi announced his result [46], Sahasrabudhe [97] achieved a breakthrough by establishing an effective inequality U (f ) ≥ c (log log log |f (1)|) 1/2−ε −1, for any f with coefficients from a finite set S ⊂ Z and effective constant c = c(S). Erdélyi [47] quickly improved 1/2 in the exponent of Sahasrabudhe's bound to 1 by combining the strengths of different methods used in [14,44,97]. For a more in-depth account refer to [48].…”