In this paper we obtain estimates for certain transcendence measures of an
entire function f. Using these estimates, we prove Bernstein, doubling and
Markov inequalities for a polynomial P(z,w) in C2 along the graph
of f. These inequalities provide, in turn, estimates for the number of zeros
of the function P(z,f(z)) in the disk of radius r, in terms of the degree
of P and of r.
Our estimates hold for arbitrary entire functions f of finite order, and
for a subsequence {nj} of degrees of polynomials. But for special classes
of functions, including the Riemann ζ-function, they hold for all degrees
and are asymptotically best possible. From this theory we derive lower
estimates for a certain algebraic measure of a set of values f(E), in terms
of the size of the set E.Comment: 40 page