  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 ${\Bbb C}^2$ 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 $\{n_j\}$ of degrees of polynomials. But for special classes
of functions, including the Riemann $\zeta$-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 $E$.
