WHEN SHOULD A POLYNOMIAL'S ROOT NEAREST TO A REAL NUMBER BE REAL ITSELF?

摘要

The conditions are studied under which the root of an integer polynomial nearest to a given real number y is real. It is proved that if a polynomial P is an element of Z[x] of degree d >= 2 satisfies vertical bar P(y)vertical bar 1/M(P)(2d-3) for some real number y, where the implied constant depends on d only, then the root of P nearest to y must be real. It is also shown that the exponent 2d-3 is best possible for d = 2, 3 and that it cannot be replaced by a number smaller than (2d - 3)d/(2d - 2) for each d >= 4.