790 research outputs found
Transcendence measures and algebraic growth of entire functions
In this paper we obtain estimates for certain transcendence measures of an
entire function . Using these estimates, we prove Bernstein, doubling and
Markov inequalities for a polynomial in along the graph
of . These inequalities provide, in turn, estimates for the number of zeros
of the function in the disk of radius , in terms of the degree
of and of .
Our estimates hold for arbitrary entire functions of finite order, and
for a subsequence of degrees of polynomials. But for special classes
of functions, including the Riemann -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 , in terms
of the size of the set .Comment: 40 page
On the complete classification of extremal log Enriques surfaces
We show that there are exactly, up to isomorphisms, seven extremal log
Enriques surfaces Z and construct all of them; among them types D_{19} and
A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the
(degree 3 or 2) canonical covering of each of these seven Z has either X_3 or
X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface
with Picard number 20 and discriminant 3 (resp. 4), which are called the most
algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by
Shioda-Inose and Vinberg).Comment: 22 pages. Math. Z. to appea
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