Transcendence measures and algebraic growth of entire functions

Dan Coman; Evgeny A. Poletsky

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arXiv · math/0403420

Transcendence measures and algebraic growth of entire functions

Dan Coman·Evgeny A. Poletsky

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Abstract

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 $ζ$-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 $E$.

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BibTeX RIS

Dan Coman, Evgeny A. Poletsky. 2004-03-24. Transcendence measures and algebraic growth of entire functions. https://arxiv.org/abs/math/0403420

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