Towards Turán's polynomial conjecture

We revisit an old problem posed by P. Turán asking whether there exists an absolute constant such that if with , then there is a polynomial with and the sum of the absolute values of the coefficients of is less than or equal to such that the sum is irreducible over . In 1970, A. Schinzel obtained a partial answer establishing the existence of an for with where is the sum of the squares of the coefficients of . Schinzel's result was further refined by P. Banerjee and M. Filaseta in 2010, obtaining with an absolute implied constant. The main contribution of the current work lies in establishing the existence of for with a significantly improved upper bound on . For instance, our main result implies the existence of an explicit for with , where the implied constant is absolute.