2 SHARP INEQUALITIES FOR THE NORM OF A FACTOR OF A POLYNOMIAL

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x)=g(x)h(x), where g and h are monic. Let \\f\\ be the maximum of \f(x)\ on the unit circle. We prove that \\g\\ less-than-or-equal-to beta(n) \\f\\ and \\g\\ \\h\\ less-than-or-equal-to delta(n) \\f\\, where beta = M(P0) = 1.38135 ..., where P0 is the polynomial P0(x,y) =1 + x + y and delta = M(P1) = 1.79162 ..., where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n --> infinity.